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# Engineering polar vortex from topologically trivial domain architecture

## Abstract

Topologically nontrivial polar structures are not only attractive for high-density data storage, but also for ultralow power microelectronics thanks to their exotic negative capacitance. The vast majority of polar structures emerging naturally in ferroelectrics, however, are topologically trivial, and there are enormous interests in artificially engineered polar structures possessing nontrivial topology. Here we demonstrate reconstruction of topologically trivial strip-like domain architecture into arrays of polar vortex in (PbTiO3)10/(SrTiO3)10 superlattice, accomplished by fabricating a cross-sectional lamella from the superlattice film. Using a combination of techniques for polarization mapping, atomic imaging, and three-dimensional structure visualization supported by phase field simulations, we reveal that the reconstruction relieves biaxial epitaxial strain in thin film into a uniaxial one in lamella, changing the subtle electrostatic and elastostatic energetics and providing the driving force for the polar vortex formation. The work establishes a realistic strategy for engineering polar topologies in otherwise ordinary ferroelectric superlattices.

## Introduction

Ferroelectric materials with spontaneous polarization have long been explored for information processing and data storage1,2,3, and one of the defining characteristics of ferroelectrics is their polar domain configurations4. Unlike their ferromagnetic counterpart, ferroelectrics usually exhibit strong anisotropy that favors particular polar axes, and polarization rotation rarely occurs5. As a result, vast majority of polar structures naturally emerging in a ferroelectric are topologically trivial, and nontrivial polar topologies such as closure-domain6,7, vortex8,9, skyrmion10, meron11, and vortex-antivortex pair12 have only recently been observed in reduced dimensions. It is now understood that in nanostructures such as nanodisks13, nanorods14, nanodots15, nanoislands16,17,18,19,20, and nanoplates21, electrostatic force dominates polar crystalline anisotropy, forcing polarization rotation and thus the formation of vortex. For ultrathin superlattice subjected to misfit strain, such polar topologies can be alternatively driven by the delicate electromechanical competitions22,23. A recent study has observed the coexistence of flux-closure and a1/a2 strip-like domain in tensile-strained PbTiO3(PTO) sandwiched between GdScO3 substrate and a SrTiO3(STO) layer, while their transformation under electron beam illumination has also been demonstrated24. These works raise the prospect of engineering topologically nontrivial polar topologies from topologically trivial domain architecture, which we seek to accomplish.

With the emergence of big data and ever increasing power consumption in microelectronics, nontrivial polar topologies become attractive as they promise alternative device configurations for high-density data storage25 as well as ultralow power negative capacitance field effect transistors26,27,28,29. From an application point of view, it is highly desirable if we can engineer nontrivial polar topologies from the ordinary ferroelectrics with topologically trivial domain architecture, and with better understanding on the energetics of polar topologies12,22, we are now in a position to explore such possibilities. Using a combination of techniques for polarization mapping30, atomic imaging6,7, and three-dimensional reciprocal space mapping (3D-RSM)31 supported by phase-field simulations32, we demonstrate the feasibility of engineering nontrivial polar topologies from topologically trivial domain architecture, and offer a realistic strategy to accomplish it. In particular, we have successfully reconstructed topologically trivial strip-like domain architecture into array of polar vortex in a model (PTO)10/(STO)10 (The subscript denotes 10-unit cell thickness) superlattices.

## Results

### Topologically trivial domain architecture of (PTO)10/(STO)10 superlattice

Guided by the previous report on strip-like domain configuration24,33,34, 17 cycles of (PTO)10/(STO)10 superlattice was synthesized on SrRuO3 (SRO)-buffered DyScO3 (001)pc (where pc refers to the pseudocubic notation) substrate using pulsed laser deposition (PLD), with details of the growth process provided in “Methods”. X-ray diffraction (XRD) data in Supplementary Fig. 1a reveals the presence of Pendellösung fringes, confirming the high quality (PTO)10/(STO)10 superlattice with a smooth interface, while atomic force microscopy (AFM) topography mapping in Supplementary Fig. 1b shows the array of terraces with ~0.4 nm-height steps, further supporting high quality of the film. Lateral piezoresponse force microscopy (LPFM) was performed to map the ferroelectric domain structure as schematically shown in Supplementary Fig. 2, which measures the shear-induced torsional displacement of the cantilever arising from in-plane polarization perpendicular to the cantilever axis. Scanning under at least two orthogonal directions is thus necessary to determine the in-plane polarization components, and we scanned by angles of 0°, 45°, and 90° between the cantilever axis and [010]pc direction. The resulted LPFM amplitude and phase mappings in Fig. 1a, b reveal a periodic strip-like domain configuration with domain wall oriented along the [110]pc direction, and no obvious contrast is seen in vertical PFM mapping over a larger area (Supplementary Fig. 3), indicating only in-plane polarization components. It is revealing to note that LPFM amplitudes scanned along 0° and 90° demonstrate opposite trend, with a peak in one direction corresponding to the valley in the other, which is evident in the line profile in Fig. 1c, while the period seen at 45° is only half of that seen at 0°- and 90°-scanning. Furthermore, 180° phase contrast is seen at 45°-scanning (Fig. 1d), while phase variation seen at 0°- and 90°-scanning is insignificant. These observations suggest alternating head-to-tail a1/a2 domain configuration as illustrated in Fig. 1e, which is topologically trivial. In particular, both a1 and a2 domains contribute to LPFM amplitude when scanned at 45°, reducing the apparent period in amplitude by half. Domain widths for both a1 and a2 domains can be estimated from the LPFM mapping over a larger area in Supplementary Fig. 3 along the line profile shown, as presented in Fig. 1f, from which the period of the strip-like domain is estimated to be 68.2 ± 2.6 nm. Such domain pattern is quite common in ferroelectrics, for example in strained PbxSr1-xTiO335 and PbTiO3 thin films36, yet it is different from the vortex array seen in (PTO)16/(STO)16 superlattice8, which we believe arises from different thicknesses. It will be exciting, however, if we can engineer such topologically trivial domain architecture into more interesting topologies, such as polar vortex.

### Emergence of polar vortex in (PTO)10/(STO)10 lamella

In order to engineer possible polar topology of (PTO)10/(STO)10 superlattice, we fabricated thin lamellas from superlattice thin film as schematically shown in Supplementary Fig. 4. For the following transmission electron microscopy (TEM) studies, we focused on a region of lamella with thickness in the range of ~20–40 nm determined using the electron energy loss spectroscopy (EELS) method37, as detailed in Supplementary Fig. 5. Low-magnification annular dark field (ADF) cross-sectional scanning transmission electron microscopy (STEM) image in Fig. 2a taken along the [010]pc zone axis reveals 17 stacking of PTO and STO layers with different contrast and approximately 8.0 nm period, and the thickness of the superlattice is measured to be 136 nm. The ADF-STEM imaging shows relatively sharp and coherent interfaces between PTO and STO, which is also confirmed by the atomically resolved energy dispersive spectrometry (EDS) mapping in Supplementary Fig. 6. Higher magnification ADF-STEM image in Supplementary Fig. 7 shows slight thickness variation in PTO layers at the unit-cell scale, though the typical configuration is determined as (PTO)10/(STO)10. We then acquired the dark field TEM (DF-TEM) image under two-beam conditions by selecting 002pc g-vector as shown in Fig. 2b, exhibiting periodic array of intensity modulation along the [100]pc direction with a spacing of ~8.6 nm. This DF-TEM image suggests a periodic distribution of clockwise-anticlockwise vortex pairs in PTO layers, as previously reported8,38,39, which is also supported by the selected area electron diffraction (SAED) pattern in Fig. 2c acquired from a region that only includes the (PTO)10/(STO)10 superlattice layers. It reveals superlattice reflections both in-plane and out-of-plane, from which the in-plane lattice parameter a and out-of-plane lattice parameter c are calculated to be ~3.9 Å and ~4.0 Å, similar to those observed in reference 4040. In particular, the enlarged view of satellite diffraction spots around (001) reflection (inset in Fig. 2c) presents the long-range order in both out-of-plane and in-plane directions. The equal space in the out-of-plane direction is measured to be ~8.0 nm reflecting the period of the (PTO)10/(STO)10 superlattices. Another set of reflections along the in-plane direction indicates long-range ordering of vortices with a period of ~8.6 nm. In order to further confirm the existence of polar vortex, the detailed atomic polar displacement was mapped by calculating the offsets between A (Pb and Sr) and B (Ti) site sublattices based on the ADF-STEM image6, and it can be clearly seen that polar vectors in the PTO layer rotate continuously around each vortex core (Fig. 2d), with alternating clockwise and anticlockwise orientations. We also calculated the corresponding tetragonality in terms of c/a ratio (Fig. 2e), wherein the red sinusoidal wave-like pattern also suggests the vortex pair as previously reported8,38,39,40. Pairs of clockwise and anticlockwise vortices are schematically shown in Fig. 2f, with their position taken from Fig. 2e, and the distance between the two adjacent anticlockwise vortices is estimated to be ~8.6 nm according to DF-TEM and SAED data. To examine the effect of substrate anisotropy, thin lamella was also cut from superlattice film in a perpendicular direction, and the resulted cross-sectional DF-TEM (supplementary Fig. 8) taken along the orthogonal [100]pc zone axis also shows periodic contrast modulation along the [010]pc direction. The corresponding electron diffraction pattern contains similar satellite spots as well, indicating the presence of periodic vortex arrays independent of lamella orientation with respect to the substrate. It is thus established that arrays of the polar vortex have been engineered in (PTO)10/(STO)10 lamellas from topologically trivial domain architecture in (PTO)10/(STO)10 superlattice film.

### 3D domain architecture of (PTO)10/(STO)10 superlattice

Piezoresponse force microscopy can only reveal polar structure near the sample surface, while 3D domain architecture can be distinct from the surface configuration. And thus there exists a possibility that polar vortex in (PTO)10/(STO)10 superlattice not recognized by PFM. To resolve this, we resort to synchrotron-based XRD as illustrated in Fig. 3a. The diffraction intensity around 002pc Bragg spot for the 17 cycles of (PTO)10/(STO)10 superlattice deposited on SRO-buffered DyScO3 was obtained via 3D-RSMs as shown in Fig. 3b, and there are no in-plane periodic satellites observed beside the PTO peak, suggesting the absence of polar vortex. As exemplified in Supplementary Fig. 9, in-plane satellite peaks always accompany the Bragg spot of the PTO reflections in (PTO)20/(STO)20 superlattice possessing periodic vortex array31. A Q[h00]-Q[00l] slice from this 002pc 3D-RSM, as shown in Fig. 3c, reveals a single alloy film peak and superlattice reflections (marked as SL (±n)). The location of alloy peak quantitatively confirms alternating a1/a2 domains in PTO with an out-of-plane lattice constant of 3.915 Å, consistent with the previous reports on a1/a2 domain structure31,40. The 15 fringes observed between superlattice peaks in Fig. 3d indicate a total of 17 cycles of (PTO)10/(STO)10, and the out-of-plane superlattice interval periodicity is measured to be 7.9 nm. Furthermore, 3D-RSM around 103pc of the DyScO3 substrate and the corresponding planar slice containing [−kk0]pc and [00l]pc are shown in Fig. 3e, f, wherein a set of significant satellites along the in-plane <110>pc are observed, corresponding to a ~70 nm periodicity of a1/a2 domains along this direction in PTO, in good agreement with PFM data in Fig. 1f. Note that the structural modulation in a1/a2 domains is completely in-plane, therefore it does not produce satellites beside any Bragg spot along the vertical [00l]pc direction. We also acquired the planar-view DF-TEM image of the (PTO)10/(STO)10 superlattice film as shown in Supplementary Fig. 10, which exhibits long-range strip-like domain along the [110]pc direction with the period estimated to be ~69 nm consistent with both PFM and 3D-RSM data. There exist a few strip-like domains with narrower periodicity, and there are a small number of strip-like domains with perpendicular orientation along [−110] as well (Supplementary Fig. 10d), which can also be observed by PFM over a larger area (Supplementary Fig. 10e). These topologically trivial domain patterns are fully consistent with continuum analysis based on strain and polarization compatibilities4, and it is distinct from the orientations and period of vortex array reported in ref. 8. These set of data thus reconfirms the topologically trivial strip-like domain structure in (PTO)10/(STO)10 superlattice before reconstruction.

### Phase-field simulation of polar vortex reconstruction

It is evident from 3D-RSM, PFM and STEM data that polar reconstruction occurs when the thin lamella is fabricated, and in order to understand the process, phase-field simulations were carried out as shown in Fig. 4. Under a biaxial strain of $${\varepsilon }_{\left[100\right]}={\varepsilon }_{\left[010\right]}=-0.3 \%$$, the superlattice forms topologically trivial strip-like domains in PTO layers consistent with PFM and 3D-RSM data presented in Figs. 1, 3. Alternating a1/a2 domains separated by $$90^\circ$$ domain walls along the [110]pc direction are found (Fig. 4a–c), while the out-of-plane polarization is negligible. When a lamella of superlattice is cut from thin film along the [010]pc direction (Fig. 4d), the biaxial strain is converted to a uniaxial one with $${\varepsilon }_{\left[010\right]}$$ released. Then nontrivial polar structure with alternating clockwise and anticlockwise vortex distribution along the [100]pc direction emerges in PTO (Fig. 4d–f), with $${P}_{\left[001\right]}$$ increasing at the expense of $${P}_{\left[010\right]}$$. It is also insightful to compare the out-of-plane lattice constant of PTO in superlattice film (3.915 Å via 3D-RSM) and lamella (4.0 Å via SAED), supporting the reconstruction of polar configuration. With out-of-plane polarization suppressed in strip-like a1/a2 domains, the corresponding lattice constant is inevitably compressed. This demonstrates that polar vortex is rather sensitive to electromechanical conditions. Indeed, by applying a larger biaxial strain of $${\varepsilon }_{\left[100\right]}={\varepsilon }_{\left[010\right]}=-1 \%$$ to the superlattice, polar vortex also emerges in superlattice film (Fig. 4g–i) without going into uniaxial lamella configuration. Here the vortex array extends along the [110]pc direction instead, and a detailed phase diagram of polar topologies is presented in Supplementary Fig. 11.

## Discussion

We can understand the reconstruction of topologically trivial domain configuration into polar vortex from energetic point of view, resulted from the competition among the electrostatic energy, the elastic energy and the gradient energy41,42. The electrostatic energy is due to the depolarization field, and when the lamella is cut from thin film, large discontinuity in polarization normal to the lamella surface is generated, preventing the out-of-lamella polarization in PTO. The remaining uniaxial strain prefers the original out-of-plane polarization, while electrostatic energy discourages it, and the competition results in the formation of vortex with delicate balance, which also increases the out-of-plane lattice constant as observed. The key lies in the reduced dimension, otherwise ordinary domain patterns can be easily formed to accommodate different requirements4. Such understanding can guide us exploring as well as engineering polar topologies further. For example, we can explore planar instead of cross-sectional lamella that may modify the magnitude of biaxial misfit strain and thus change the energy landscape, which is currently under investigation.

With respect to potential applications, the polar vortex has been suggested for high-density data storage, yet in a thin film configuration, the vortex structure is usually one-dimensional in nature, as exhibited by Fig. 4g–i, which reduces the storage density substantially. Our lamella configuration overcomes such difficulty by accommodating arrays of zero-dimensional polar vortex, as exhibited by Fig. 4d–f. Other applications such as negative capacitance field effect transistors can also be envisioned. On more fundamental side, we demonstrate that such polar vortex can be engineered from topologically trivial domain configuration, expanding the reach of topological polar structures into the otherwise ordinary ferroelectric superlattice. In this regard, our work also highlights the importance of combining surface probing, advanced microscopies, and 3D structural characterization to accurately determine the polar topologies. Reconstruction can occur, and thus information from a single microscopy technique could be incomplete.

## Methods

### Fabrication of (PTO)10/(STO)10 superlattices

Superlattices of (PTO)10/(STO)10 (where 10 denote the thickness of alternating stacks of the STO and PTO layers in term of unit cells) were fabricated on 12 nm SRO-buffered (110)-DyScO3 substrates in a pulsed laser deposition system (PVD-5000) equipped with a KrF excimer laser (λ = 248 nm). The SRO-buffered layer was first grown at a 690 °C substrate temperature and 80 mTorr oxygen pressure. Then, the substrate temperature was cooled to 600 °C and the oxygen pressure went up to 200 mTorr for the subsequent alternating growth of the PTO and STO layers. The laser energy for the growth of the SRO and the (PTO)10/(STO)10 superlattices was 400 and 360 mJ pulse−1, respectively. Selecting the correct laser energy was crucial for fully strained PTO and STO sublayers, which is a prerequisite for the presence of polar vortices in the superlattice. Thicknesses of the SRO, PTO, and STO layers were held to be 30-, 10-, and 10-unit cells, respectively, by controlling the growth time. After growth, the (PTO)10/(STO)10 superlattice films were slowly cooled to room temperature at a 200-mTorr oxygen pressure.

### Atomic force and piezoresponse force microscopies

The (PTO)10/(STO)10 superlattices were characterized using a commercial scanning probe microscope (Asylum Research, MFP-3D infinity) in tapping mode to access the morphology, and in the dual AC resonance tracking (DART) piezoresponse mode (both in-plane and out-of-plane modes) to record the ferroelectric domain structure. Pt conductive tips with a force constant of 10 N m−1 (RMN-25PT400B) were used for piezoresponse force microscopy measurement. Since in-plane (IP) PFM only measures the component perpendicular to the cantilever axis, which is associated with the torsional vibration of the cantilever, we gathered the real part images of IP PFM signals measured at different probe orientations to construct a full vector map of the IP piezoresponse vector.

### Lamella preparation

The planar-view and cross-sectional lamella specimens were thinned to ~30 μm by using mechanical polishing or cut via focused ion beam (FIB), and then by performing argon ion milling. The argon ion milling was carried out using the Precision Ion Polishing System with accelerating voltage of 3.5 kV and then stepped down to a final cleaning voltage of 0.3 kV.

### Dark field and ADF images

Diffraction contrast TEM experiments were carried out using JEOL ARM 300 F instrument at 300 kV. Dark field imaging was performed in the two-beam condition with the samples being tilted off the zone axis and imaged in the dark field using 002pcg vector. ADF-STEM images were recorded from an aberration-corrected FEI Titan Themis G2 at 300 kV. The convergence semi-angle for imaging was 30 mrad, and the collection semi-angle was 39–200 mrad. The atom positions were determined by simultaneously fitting two-dimensional Gaussian peaks to an a priori perovskite unit cell using a MATLAB code. The polar vectors were plotted from the offset between A (Pb and Sr) and B-site (Ti) sublattices based on the ADF-STEM image in Fig. 2d. We acquired the ADF-STEM images with a smaller convergence semi-angle for enhancing the stress contrast in the strained PTO layer. Different convergence semi-angle will only affect the relative intensity of the atomic column, not the position of the atom columns.

### Thickness measurement of the lamella

The lamella thickness was estimated using the STEM-EELS method. STEM-EELS was performed using an aberration-corrected FEI Titan Themis G2 at 300 kV. The typical energy resolution (half-width of the full zero-loss peak, ZLP) was 0.8 eV. For the different regions of superlattice lamella, STEM-EELS images in the low loss region were acquired and converted to a thickness map of inelastic mean free paths (t/λ) in Gatan’s Digital Micrograph software using the absolute log-ratio method for the zero-loss and the first plasmon peaks37.

### Synchrotron-based X-ray diffraction

The 3D-RSM obtain from synchrotron X-ray diffraction were employed to study the complex phase coexistence in the superlattice structures. The 3D-RSM experiments were performed at the 33-BM-C beamline of the Advanced Photon Source and BL14B beamline of Shanghai Synchrotron Radiation Facility. High accurate 3D-RSMs were obtained by the excellent accuracy of Huber 6-circle diffractometer equipped with Pilatus 100K or Eiger X 500K area detector.

### Phase-field simulation

In the phase-field model, the free-energy density $$f$$ of the STO/PTO superlattice has the following form

$$f=\; {\alpha }_{i}{P}_{i}^{2}+{\alpha }_{{ij}}{P}_{i}^{2}{P}_{j}^{2}+{\alpha }_{{ijk}}{P}_{i}^{2}{P}_{j}^{2}{P}_{k}^{2}+\frac{1}{2}{c}_{{ijkl}}{\varepsilon }_{{ij}}{\varepsilon }_{{kl}}-{q}_{{ijkl}}{\varepsilon }_{{ij}}{P}_{k}{P}_{l}\\ +\frac{1}{2}{g}_{{ijkl}}{P}_{i,j}{P}_{k,l}-\frac{1}{2}{\varepsilon }_{0}{\varepsilon }_{r}{E}_{i}{E}_{i}-{E}_{i}{P}_{i},$$
(1)

in which $${\alpha }_{i}$$, $${\alpha }_{{ij}}$$ and $${\alpha }_{{ijk}}$$ denote the Landau expansion coefficients, $${c}_{{ijkl}}$$ are the elastic constants, $${q}_{{ijkl}}$$ are the electrostrictive coefficients, $${g}_{{ijkl}}$$ are the gradient coefficients, $${\varepsilon }_{0}$$ is the vacuum permittivity and $${\varepsilon }_{r}$$ represents the relative dielectric coefficien. $${P}_{i}$$, $${\varepsilon }_{{ij}}$$ and $${E}_{i}$$ are the polarization, strain and electric field components, respectively. All repeating subscripts in Eq. (1) imply summation over the Cartesian coordinate components xi (i = 1, 2 and 3), and subscript “,$$i$$” denotes the partial derivative operator with respect to $${x}_{i}$$ ($$\partial /\partial {x}_{i}$$).

The temporal evolution of the polarization and domain structure can be described by the time-dependent Ginzburg–Landau (TDGL) equation as

$$\frac{\partial {P}_{i}\left({{{{{\boldsymbol{r}}}}}},t\right)}{\partial t}=-L\frac{\delta F}{\delta {P}_{i}\left({{{{{\boldsymbol{r}}}}}},t\right)}$$
(2)

where $$L$$ represents the kinetic coefficient that controls the speed of domain evolution; $$F=\int _{V}f{{{{{\rm{d}}}}}}V$$ is the total free energy in the whole structure; $${{{{{\boldsymbol{r}}}}}}$$ and $$t$$ denote the spatial position vector and time, respectively. In addition, both the mechanical equilibrium equation

$${\sigma }_{{ij},j}=\frac{\partial }{\partial {x}_{j}}\left(\frac{\partial f}{\partial {\varepsilon }_{{ij}}}\right)=0,$$
(3)

and Maxwell’s equation

$${D}_{i,i}=-\frac{\partial }{\partial {x}_{i}}\left(\frac{\partial f}{\partial {E}_{i}}\right)=0,$$
(4)

are satisfied simultaneously for a body-force-free and charge-free ferroelectric system, where $${\sigma }_{{ij}}$$ and the $${D}_{i}$$ are the stress and electric displacement component, respectively. The semi-implicit Fourier-spectral method32 is employed to solve the partial differential equation. Three-dimensional simulation is carried out for clearer illustration and computational simplicity, where $$3\times 3$$ periodic PTO/STO superlattices model is adopted here. And $$150\Delta x\times 150\Delta x\times 60\Delta x$$ grid point with $$\Delta x=0.4\;{{{{{\rm{nm}}}}}}$$ in real space is used for space discretization, where 10-grid is used along the thickness direction of each PTO and STO layer. And the length of time step is chosen as $$\Delta t/{t}_{0}=0.2$$, where $${t}_{0}=1/({\alpha }_{0}L)$$ with $${\alpha }_{0}$$ representing the absolute value of $${\alpha }_{1}$$ at room temperature. Periodic boundary conditions for electric potential and polarization components are employed along the [100]pc, [010]pc, and [001]pc direction, respectively. The material parameters for PTO and STO used in the simulation are listed in Table 1. Small random fluctuation (<0.01$${P}_{0}$$, where $${P}_{0}=0.76$$ C m−2 is the spontaneous polarization of PTO at room temperature) is used as the initial setup for polarization to initiate the simulation.

## Data availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors.

## Code availability

The MATLAB-based toolbox for fitting atom positions and calculating local polarization is available from the corresponding author upon request.

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## Acknowledgements

This work was supported by the National Key R&D Program of China (Grant Nos. 2016YFA0201001, 2016YFA0300102), the National Natural Science Foundation of China (Grant Nos. 51872251, 11875229, 11972320, 11675179, 11627801, and 92066203), the National Equipment Program of China (ZDYZ2015-1), the Key R&D Program of Guangdong Province (Grant Nos. 2018B030327001, 2018B010109009, and 2019B010931001), Shenzhen Science and Technology Innovation Committee (Grant Nos. JCYJ20170818163902553, JCYJ20200109115219157), the Leading Talents Program of Guangdong Province (2016LJ06C372), Guangdong Provincial Key Laboratory Program (2021B1212040001) from the Department of Science and Technology of Guangdong Province, Zhejiang Provincial Natural Science Foundation (Grant No. LZ17A020001), and the “2011 Program” Peking-Tsinghua-IOP Collaborative Innovation Center for Quantum Matter. We also acknowledge the Electron Microscopy Laboratory in Peking University for the use of the Cs-corrected electron microscope. Z.L. acknowledges the support from the beamline BL14B of Shanghai Synchrotron Radiation Facility (SSRF) and the beamline 33BM of Advanced Photon Source.

## Author information

Authors

### Contributions

J.L., C.T., P.G., J.W. and Z.L. conceived the idea and designed the work. C.T. grew the samples assisted by J.B.W., X.Z., M.L. and X.O; Y.S., C.P. and R.Z performed the electron microscopy experiments and assisted by M.W., K.L., X.B., D.Y. and J.Z. under the direction of P.G.; Y.D. and Z.L. carried out synchrotron structural characterization; C.L. did the phase-field simulation under the direction of J.W. and J.L.; C.T. and Y.S. performed data analysis assisted by Y.D. and C.L. under the direction of J.L., J.W., Z.L. and P.G.; C.T. and J.L. wrote the manuscript with the assistance of Y.W.S., Y.D. and C.L. and all the authors. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Jie Wang or Peng Gao or Zhenlin Luo or Jiangyu Li.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Takao Matsumoto, Xiuliang Ma and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Tan, C., Dong, Y., Sun, Y. et al. Engineering polar vortex from topologically trivial domain architecture. Nat Commun 12, 4620 (2021). https://doi.org/10.1038/s41467-021-24922-y

• Accepted:

• Published:


text:
# What's the maximum (average) damage that can be dealt in one round by a 20th level character with at least two levels of Fighter?

What is the most effective way to use a level 20 character's (with two or more levels in fighter for Action Surge ability) in order to maximize single target damage done in one round?

Rules/Restrictions:

• Any race is allowed.
• If there is a circumstantial bonus, such as Assassinate or Haste, please show each step in damage calculation
• No magic items, charms, blessings, or epic boons.
• No, the "damage done" can't be to yourself.
• Vulnerability Damage caused by another player's action should not be considered.
• Rogues may consider themselves to be targeting a Surprised creature.
• You do not have advantage by default. If you can generate some advantage for yourself, then you may do so.
• Consider your target to have AC20; Saving Throws Str +5, Dex +6, Con+5, Int +1, Wis +9, Cha +10; Skill Checks str(+5), dex(0), con(+5), int(+1), wis(+3), cha(+4); medium size... not dissimilar to a Death Knight.
• You may have one non-combat round of setup.

(Again. Yes, I know this is alpha damage / nova damage and isn't sustainable.)

• Comments are not for extended discussion; this conversation has been moved to chat. – mxyzplk Feb 25 '18 at 7:06

## Expected Average Damage: 641.354, Maximum Damage: 1185

No items or spell components, no special circumstances. Legal for use in Adventurer's League. Attacking with Surprise.

Half-Orc (Arthur) (Savage Attacks add an extra die of damage on every critical hit)

Divine Soul Sorcerer 4 (Favored by the Gods, Shadow Blade spell, Subtle Spell meta-magic, ASI)

War Domain Cleric 1 (War Priest - make one weapon attack as a Bonus Action)

Assassin Rogue 3 (Assassinate - if surprised, advantage on all attacks, critical on all hits, 2d6 Sneak Attack with Finesse weapons per turn, Expertise: Stealth, Perception)

Battle Master Fighter 3 (Action Surge, 4 Maneuvers, including Precision Attack, Menacing Attack, and Disarming Attack, Fighting Style Dueling)

Paladin 2 (Divine Smite, Divine Sense, Fighting Style Armored)

Gloom Stalker Ranger 5 (Dread Ambusher: in the first round you gain an extra attack on every Attack action in the first round, +10 movement, extra 1d8 damage of the weapon's type on this extra attack, Primeval Awareness, Pass without a Trace, Longstrider, ASI, Fighting Style Archery)

Strength 14 + 2 + 2 ASI

Dexterity 13

Constitution 8 + 1

Intelligence 13

Wisdom 13

Charisma 13

• To-Hit: +11, +12-19 with Precision Attack
• Hit Points: 92
• AC: 19 (Breastplate, Armored Fighting Style, Shield)
• Stealth: +23 (Expertise and temporary boost from Pass without a Trace)
• Initiative: +2 (+1 Dexterity, +1 Wisdom from Dread Ambusher)
• Perception: +13 (Expertise)
• Level 10 spellcaster (4 Sorcerer + 1 Cleric + (2 Paladin + 5 Ranger)/2 + 2 Wizard)
• Movement speed: 40 feet (50 feet with Longstrider)
• Darkvision: 90ft

The crux of it is that you have created a truly dreadful ambush character which has significant precision-boosting skills and tracking skills for his enemies. Beyond just being able to hit things hard, they also are able to do so with uncanny accuracy. This character goes patrolling for evil and predators in the woods, using his Primeval Awareness (L1 spell slot) to identify that there is evil in his land. As he uses his hunting ability to stalk his prey, his high perception and natural stealth make it difficult for his quarry to hide and make himself similarly difficult if not nearly impossible to spot. He'd use Pass Without a Trace, perhaps subtly, so that he could sneak up on his quarry. As he draws near and identifies his target, he casts Longstrider from his Ranger spells using the Subtle Spell meta-magic just to give himself that extra boost in movement that he'll need in order to close with his quarry before it realizes the danger. He knows how long it will take him to get into striking range, and stealthily moves to within 50 feet of his target. He has foreseen this day's events, and knows that death awaits his enemy. Patiently, he waits until it's his turn to strike. In terms of the sequence of events, I imagine it would look like this:

Primeval Awareness of the quarry, Subtle Pass without a Trace + Subtle Longstrider during the hunt for the quary. In the preparation round, cast Subtle Shadow Blade L5, from which the surprise round begins. Portent determines the initiative order for the target and the attacker: + Assassinating Critical: Action Surge + Dueling Attacks + Divine Smites + Dread Ambusher + Sneak Attack + Bonus Action War Priest weapon attack with Divine Smite.

# Resource Expenditure:

• Spending 2L5, 3L4, 3L3, Spell Slots, all maneuvers, action surge, divine favor, War Priest ability
• Optional for flavor Spend another L2, 2L1 Spell slots: use a level 1 spell slot for Primeval Awareness to identify that your enemy exists nearby, then cast Pass without a Trace and Longstrider using Subtle Spell to hunt them and ambush them successfully.

# Order of Events:

Preparation Round: Level 5 Shadow Blade. (4d8 psychic damage per melee attack)

Surprise Round: Initiative: Use Portent rolls to guarantee Assassinate feature works. Action: Attack action (2 attacks from Ranger, +1 attack from Dread Ambusher feature) *Bonus Action: 1 attack through War Priest Action Surge: Attack action (2 attacks from Ranger, +1 attack from Dread Ambusher feature)

## The Math on the Damage:

(All dice are doubled because these are critical hits because of Assassinate)

Every attack that hits results in this: 2$$\\times\$$4d8 Psychic (Shadow Blade, Level 5) + 1d8 Savage Attack + 2 Dueling Fighting Style + 5 Strength modifier = 9d8 + 7 Psychic damage

7 attacks (3 attacks per action, plus 1 attack from War Priest) yield: 63d8 + 49 Psychic damage

Gloom Stalker Ranger allows us to adds 1d8 damage of the original weapon's damage type to the extra attack of every Attack action we take in the first round of combat through the Gloom Stalker feature. Since we take two Attack actions through Action Surge, we get extra 2d8 Psychic damage, which is doubled since they're critical hits. 67d8 + 49 Psychic damage

Assassin Rogue allows us to add 2d6 Sneak Attack damage because Shadow Blade is a Finesse weapon. Sneak attack rules for rogues do not require you to use Dexterity when using Finesse weapons. Since the attack is a critical hit, the damage gets doubled. 67d8 + 4d6 + 49 Psychic

As a paladin, every successful hit allows us to add 2d8 + (spell slot level-1)$$\\times\$$1d8 radiant damage, maxing out at 5d8 for an L4 spell slot or higher. As a 10th level spellcaster, after casting Shadow Blade at level 5, we can use 3 L4 spell slots, 1 L5 spell slot, and 3 L3 spell slots. All of this, of course, gets doubled on critical hits, which these will be. This results in the following additional damage: 2$$\\times\$$(3$$\\times\$$5d8 + 1$$\\times\$$5d8 + 3$$\\times\$$4d8) = 64d8 Radiant damage

## Damage Total:

(67d8 + 4d6 + 49 Psychic) + (64d8 Radiant) = (67$$\\times\$$4.5 + 4$$\\times\$$3.5 + 49 Psychic) + (64$$\\times\$$4.5 Radiant) = 364.5 Psychic + 288 Radiant = 652.5 total

In order for the Assassinate feature to work, during the first round of combat the Assassin must act before their quarry. By having access to Portent, we will have two dice rolls which can be used for any ability check, which includes initiative. Because the quarry has a +0 to their initiative and our character has a +2 to initiative, regardless of what we roll, we can substitute the lower of the two Portent rolls to the target and use the higher of the two rolls for ourselves.

The following probability calculations below were made with the help of the table in the article "D&D 5e: Probabilities for Advantage and Disadvantage" to figure out the likelihood of rolling particular numbers on a 20-sided die with advantage.

At least one attack will always hit. It's true that it's mathematically possible for a 1 to be rolled on every single attack die (7 attacks, all with advantage), but the odds of that are ridiculously low: $$\\left(\left(\frac{1}{20}\right)^2\right)^7\$$ = 0.0000000000000000006103515625. In other words, the Sneak Attack will always be added.

Precision Attack maneuvers and the Favored by the Gods feature allow the player to use them after the attack roll is made and any of these three abilities can be used in conjunction with one another.

With our target having an AC of 20, and our character having a starting +11 to hit, we need to roll 9 or higher in order to successfully land a hit, which from the table we can see that we have an 84% likelihood through advantage.

The Precision Attack maneuver from the Battle Master, adds 1d8 to attack rolls, yielding 4.5 average increase in likelihood. This can be used on 4 attacks, resulting in +15.5 to attack rolls, since we have 4 superiority dice from the Battle Master. In addition, the Favored by the Gods Divine Soul Sorcerer feature will allow us to add 2d4 to one attack that misses, turning it into a hit on average, due to effectively adding +5 to hit (resulting in a +16 to hit) for one of the two attacks which we cannot add the Battle Master die to. As a result, for 1 attack we need to roll 4 or higher through advantage to score a hit, and for 4 of the other 5 attacks we need to roll 4.5 or higher through advantage in order to score a hit.

The optimal strategy of their use should look something like this:

1. Our attacks, whenever possible, should be sequenced from most damaging to least damaging, in order to take advantage of our ability to lazily add attack bonuses.
2. Similarly, we should use our spell slots for Divine Smite in a decreasing progression. The Bonus Action attack will use the L5 spell slot, the first three attacks of the first Attack action will each use an L4 spell slot, and the three attacks of the second Attack action will each use an L3 spell slot.
3. If an attack misses and we still have a superiority die, we should use the Precision Attack adds an average of 4.5 to the attack roll.
4. If the total for the attack roll is still too low and we still can use Favored by the Gods, we should use it.
5. If the number of attacks remaining is less than the number of superiority dice remaining and we haven't applied a maneuver to the attack, we should apply a damaging maneuver like Menacing Attack or Disarming Attack.

The Bonus Action weapon attack will average (2$$\\times\$$4d8 + 1d8 + 7) Psychic + 2$$\\times\$$5d8 Radiant = 47.5 Psychic + 45 Radiant = 92.5 damage. Adjusted for the likelihood of a natural 1, the accuracy adjustment reduces the expected damage total by: -92.5*(1-0.84) = -14.8

When using Favored by the Gods, our to hit becomes +16 on average, which means we need to roll 4 or higher through advantage. On the referenced table, we see that for this we have a 97.8% likelihood. The damage for this attack will be (2$$\\times\$$4d8 + 1d8 + 7) Psychic damage and (2$$\\times\$$5d8) Radiant = 9d8 + 7 Psychic + 10d8 Radiant = 47.5 Psychic + 45 Radiant = 92.5 damage. Adjusted for accuracy, this means that the damage total is lessened by -(1-0.978)$$\\times\$$92.5 = -2.035.

When using the Precision Attack maneuver, since the AC of the target is 20 and we will need to roll "5.5", the average (I think) will be (97.8% + 96%)/2 = 96.9%. This means that each of the 4 attacks with maneuvers has a 96.9% chance to hit. So, we will on average do 96.9% of the expected damage for those 4 attacks. The damage of these 4 attacks of the two Attack actions is: 4$$\\times\$$1d8 + 2$$\\times\$$4$$\\times\$$4d8 + 2$$\\times\$$1d8 + 4$$\\times\$$7 Psychic + 2$$\\times\$$(4d8 + 4d8 + 5d8 + 5d8) Radiant = (38d8 + 28) Psychic + (36d8 Radiant) = 199 Psychic + 162 Radiant = 361 total damage. Adjusting for accuracy, this means that the previously mentioned "total damage" is also lessened by -361$$\\times\$$ (1 - 0.969) = -11.191

On each of the 4 attacks on which we use a superiority die, if the attack doesn't need the accuracy boost, we can apply the die to the damage by using a different maneuver, such as Disarming Attack, doubling it because of the critical hit. This results in 2$$\\times\$$1d8 damage = 9 damage. How often would we get to add this extra damage? Since the original to-hit modifier is +11, hitting AC 20 would happen 84% of the time. Thus, 84% of the time, we can substitute a damaging maneuver instead of the Precision Attack Battle Master maneuver. Thus, the expected damage is increased by: 0.84$$\\times\$$9 onto every attack, for four attacks: 0.84$$\\times\$$9$$\\times\$$4 = +30.24

The 3rd attack from the second Attack can do (2$$\\times\$$4d8 + 1d8 + 7) Psychic damage and (2$$\\times\$$4d8) Radiant = 9d8 + 7 Psychic + 8d8 Radiant = 47.5 Psychic + 36 Radiant = 83.5 damage. As such, the damage expected for the last attack will be 83.5*(1-0.84) = -13.36.

# Result:

Adjusted for accuracy, then, this means that the expected average damage of this character is 652.5 -14.8 - 2.035 -11.191 + 30.24 -13.36 = 641.354 total damage

Hope this math was relatively easy to follow, and hope you guys enjoyed the character and story setup as well. And be aware, the optimal use of the superiority dice and Favored by the Gods could improve this by a few more points.

*Note: Originally, I came up with a slightly higher number by using a Quickened Booming Blade. However, that spell is from Sword Coast Adventurer's Guide, and because many of the other features referenced here are from Xanathar's Guide to Everything, in Adventurer's League we wouldn't be able to use both for this character. *

It's worth noting, since the original poster mentioned this looking like a Death Knight, that this build would probably work even better against a Death Knight, because the latter's shield and sword can be removed through the Disarming Attack maneuver, thus lowering his AC and removing its ability to parry with his reaction.

*Note: Another poster, Herohersh (https://rpg.stackexchange.com/a/138689/49041), cleverly suggested that based on the guidelines for this math problem you could use your action to Create Bonfire during the setup round when you're casting Shadow Blade, step into the bonfire in order to trigger Absorb Elements through your reaction, and then proceed to attack using the bonus damage of Absorb Elements. In addition to probably not working mechanically due to stipulations regarding casting any other spell during the same round during which a Bonus Action spell is cast, for my personal taste I find this unsatisfying because it breaks the immersion and idea that you could still be hidden from your enemy the round before you attack if you conjure a bonfire that gives away your position. *

• This looks great! I'd love of another math savvy user could triple check your math, but I'll likely change the accepted answer sometime this week. – goodguy5 Jul 21 '19 at 13:08
• Math seems to check out. I got slightly different numbers (off by about 20), but I may have implemented the accuracy boosting differently. Bravo! [although don't think I won't be back to try to reclaim my throne ;)] – David Coffron Jul 22 '19 at 15:56
• I get an average damage of 655.240 based on my methodology and a maximum damage of 1185. I'm not too worried about the difference in average since that mostly comes down to methodological differences and is pretty close to what you estimate, but I can't account for 24 extra points of damage in your calculated maximum. It appears that you've accidentally doubled the Sneak Attack dice twice, instead of just once. – Xirema Jul 23 '19 at 16:06
• I added some MathJax to make some of your maths easier to read. Feel free to call me on it if I messed up any of the equations. – Someone_Evil Jul 23 '19 at 20:24
• The portent feature can only be used once per turn. While rolling initiative is before the rounds start, I think it is likely to be considered simultaneously so you might only be able to replace the role of one person. – findusl Nov 8 '19 at 8:14

Since the question states most effective way to deal max damage, I inferred that expected damage, which includes hit chance, critical hit chance, and saving throws, is the best metric (I have another answer for assuming perfect rolls, but that would be another question), so...

# Expected damage is 564. Maximum is 648. Here's how:

## The Character

Chloe is a level 20 Goblin {ASs: 9, 18, 10, 20, 8, 14} with these classes...

• Wizard 17 ---------- [for Spells, Empowered Evocation, Alert feat and 3 ASI]
• Fighter 2 ----------- [for Action Surge]
• Warlock 1 ---------- [for Hexblade's Curse]

## The Preparation

1. Chloe casts wish replicating the effect of hallow with Energy Vulnerability on force damage within 60 feet of the target.

## The Round

2. Chloe casts magic missle (level 8 spell slot) applying Fury of the Small.
3. Chloe uses Action Surge.
4. Chloe casts magic missle (level 7 spell slot).

## The Calculation

• All force damage is doubled from hallow's Energy Vulnerability...
• The target never enters the area
• Chance target goes first: 15.125% (using optional rule for ties to avoid DM fiat)
• Chance to save: +10 vs 19 DC = 60%
• totals to 1 - (0.15125*0.6) = 90.925% chance
• Each dart from the first magic missle deals...
• 1d4 + 1 + 5 (Empowered Evocation) + 6 (Hexblade's Curse) force damage
• 10 darts, doubled 90.925% of the time = 19.90925 multiplier
• Expected: (2.5 + 12)*19.0925 = 276.84125 [Max: 320]
• Each dart from the second magic missle deals...
• 1d4 + 1 + 5 (Empowered Evocation) + 6 (Hexblade's Curse) force damage
• 9 darts, doubled 90.925% of the time = 17.18325 multiplier
• Expected: (2.5 + 12)*17.18325 = 249.157125 [Max: 288]
• Fury of the Small deals...
• 1 use per short rest, doubled 90.925% of the time = 1.90925 multiplier
• Expected: 20*1.90925 = 38.185 [Max: 40]

Total expected: 564.183375 [Max: 648]

Note: This assumes that the target chooses to make the Charisma save against hallow; it doesn't have to. If not, the expected damage increases to 591

Rulings

How does Empowered Evocation work with Magic Missile?

• Is this allowed RAW? Pretty sure there's a rule against casting two non-cantrip spells in the same turn. – Nathan Sep 11 '18 at 18:01
• @Nathan it is allowed. That rule only applies to bonus action spells. Chloe is casting 2 action spells here. See this question for more details. – David Coffron Sep 11 '18 at 18:06
• Creature is a Tarrasque...wizards insta kills itself if the magic missile spell is reflected back – illustro Apr 13 '19 at 18:25
• @dsollen Even if the target is surprised, he still gets a turn in the first round. There is no "surprise round" in 5e where the surprised targets don't get a turn; they just don't get to act on the turn they are surprised. – David Coffron Jul 22 '19 at 20:55
• @findusl even if a creature doesn't act on their turn, they still make a save at the end of their turn. The surprised state in 5e just makes it so they can't take actions or reactions until the end of their first turn. – David Coffron Nov 8 '19 at 12:22

# 249 expected, or 433 max damage

We are using a Scourge Aasimar Shadow Magic Sorcerer 18/Fighter 2, with a Charisma score of 20

The round before, activate Radiant Consumption. On your turn, move to within 5 feet of the target, use a bonus action to summon your Hound of Ill Omen next to the target, cast Disintegrate, action surge -> cast Meteor Swarm.

## Scourge Aasimar Contribution:

• At the end of each turn, you and each creature within 10ft of you take radiant damage equal to half your level: 10

• Once on each of your turns, you can deal extra radiant damage to one target when you deal damage to it with an attack or a spell. The extra radiant damage equals your level: 20

Damage so far: 30

## Fighter Contribution:

Action Surge

Damage so far: 30

## Sorcerer Contribution:

Hound of Ill Omen:

• Uses the statistics of a Dire Wolf
• While the hound is within 5 feet of the target, the target has disadvantage on saving throws against any spell you cast.
• On the hound's turn (it rolls initiative separately) it attacks.
• The hound has +5 to hit and deals 2d6+3 damage. It has advantage if an ally is within 5 feet of the target (which you are).
• Vs an AC of 20 the hound has a 49% chance of hitting with advantage, and deals 10 expected damage (max of 15, or 27 max on crit)

Damage so far: 40 (57 max), and target has disadvantage on its saving throws vs spells you cast.

Disintegrate:

• 10d6+40 force damage, +3d6 for each level slot above 6th spent. Dex save negates
• Cast using an 8th level slot, for 16d6+40 damage
• Average damage is 16 * 3.5 + 40 = 96
• Saving throw DC = 8 + proficiency + Cha mod = 8 + 6 + 5 = 19
• Chance of making the ST = 40% (Target has +6 bonus).
• Chance of making the ST with disadvantage = .4 * .4 = 16%
• Expected damage = average damage * P(failing saving throw) = 96 * .84 = 80.64 (136 max)

Damage so far: 120.64 (193 max)

Meteor Swarm:

• 20d6 Fire, 20d6 Bludgeoning, Dex save for half
• Average damage is 40 * 3.5 = 140
• Saving throw DC = 8 + proficiency + Cha mod = 8 + 6 + 5 = 19
• Chance of making the ST = 40% (Target has +6 bonus).
• Chance of making the ST with disadvantage = .4 * .4 = 16%
• Expected damage = average damage * P(failing saving throw) + average damage/2 * P(making saving throw) = 140 * .84 + 70 * .16 = 117.6 + 11.2 = 128.4 (240 max)

Damage so far: 249 (433 max)

PS: You could get slightly higher damage using the Empowered Metamagic, but that complicates the calculations significantly. After some exhausting-looking calculations in chat and in a question, Empower will add ~12 expected damage to Meteor Swarm, and ~10 to Disintegrate, for ~22 more expected total.

PS2: The original specified target was a Death Knight, AC20, +6 to Dex saving throws.

PS3: If you are trying to find max damage (with all perfect rolls) you could swap Disintegrate for Scorching Ray, which brings the total up to 456 but only if every ray crits.

• Comments are not for extended discussion; this conversation has been moved to chat. – mxyzplk Feb 25 '18 at 7:06

## Average 396, Max 1030

High Elf Eldritch Knight 12/Whispers Bard 5/Paladin 2/Hexblade Warlock 1, with Dexterity 20 and the Elven Accuracy, Lucky, Martial Adept and Savage Attacker feats.

Prep round: Bonus action: cast a 5th level Shadow Blade. Action: Apply purple worm poison to our blade.

Go Time: Bonus action: Apply Hexblade's Curse to our foe.

Action, Surge: We are going to make 6 weapon attacks. That is what is happening here.

To-hit: Assume dim light (given we can assume surprise, this seems fair)*. Now we have super elf-advantage on all of our attacks, giving us a to-hit of 94% and a crit chance of 27%. On any attacks we would miss, we apply Lucky, giving us a 60% chance of turning a hit into a miss, for a total to-hit chance of around 97%. (Given there is a 0.05% chance of missing three or more times, I will assume I will need at most two uses of it here.) We will also apply Lucky once our number of attacks left is equal to our number of Lucky uses left, for more crit chances.

On hit: Our sword does 4d8+13 (Dex 20, Hexblade's Curse, Dueling Fighting Style) damage, and we can pour all our high-level spell slots into Divine Smites. We have 4 4th-level or higher (5d8) and 2 3rd-level (4d8) slots to burn through. Finally, on one of our attacks (our first crit) we can use Bardic Inspiration to do an extra 3d6 damage. Thus, if all 6 attacks hit, we are dealing 52d8 + 3d6 + 78 damage as a baseline. This is average 270, but also slightly optimistic.

Critting: Our first crit, if it lands, will deal an extra 9d8 + 3d6 damage; we can also use Savage Attacker to reroll the 8d8 from our weapon, for a total average damage increase of 55. Our second, third and fourth crits, without Savage Attacker or Inspiration, do 41 on average. Subsequent crits will average 36 due to the lower level spell slots. This ends up giving us a total of 86 average damage.

Missing: Missing is interesting, because in order to maximise our crit damage, we have to save our high-level spell slots for later hits in case they crit. This means a miss is usually (and for the purposes of this model, always) one of our biggest attacks, losing us 53 damage. Of course, between a static miss chance of 97%, and a Precision Attack d6 that can save us from a miss with probability 81%, this is not a big difference. The chance of us missing once is 4%; the chance of us missing twice is negligible. That works out to about 2 lost damage.

Max: The maximum is just the maximum of 6 crits, which comes out to the max of 104d8 + 6d6 + 78, or 946.

Trip Attack: If all 6 attacks hit without needing our Precision Attack, we make our last attack a Trip Attack, dealing an extra 1d6 damage. This adds 3 to our average, and 12 to our max.

Poison: One of our attacks will hit. Then the purple worm poison kicks in, dealing 12d6 damage on a failed save or half on a success. If we use our third dose of Lucky here, our foe succeeds 12% of the time, so our poison does Average 39, max 72

Totals:

Average damage: 396

Max damage: 1030

*If we must have bright light, we can swap our Warlock level for an extra level of Paladin for Vow of Emnity, which gets us the advantage but not the other perks.

# Average: 308.866 — Max: 470

• Race: Protector Aasimar
• Class Levels: Fighter (Battle Master) 3/Rogue (Assassin) 17
• Feats: Crossbow Expert, Sharpshooter
• Equipment: 2 Hand Crossbow, 3 doses of Purple Worm Poison (nonmagical) on 3 crossbow bolts.
• Relevant ability scores: Dex +5
• Fighting Style: Archery.

# Step by step:

Previous turn

• Radiant Soul: Your transformation lasts for 1 minute or until you end it as a bonus action. During it, you have a flying speed of 30 feet, and once on each of your turns, you can deal extra radiant damage to one target when you deal damage to it with an attack or a spell. The extra radiant damage equals your level (20).

PS: Why not Scourge Aasimar? Well, being an assassin and radiating bright light doesn't seem to be very functional since the character relies on surprise her foes.

This turn

• Attack the surprised target with a poisoned bolt from the hand crossbow using Sharpshooter. Thanks to Assassinate, is a critical hit and thanks to Death Strike, it doubles the damage.

1d6(H.Crossbow) + 10(Sharpshooter) + 5(Dex) = 1d6+15

Chance to hit with Precision Strike: 1 - Chance to Miss: 1 - 35% * 35% = 87.75%

Critical: 2d6+15 -> Avg: 22 / Max: 27 -> Times 2 (Death Strike: chance is 65%)

Partial Total (multiplied by 87.75% and chance of death strike): Avg: 31.853 / Max: 54

• Bonus Action: Attack the surprised target with a poisoned bolt from the other hand crossbow using Sharpshooter. Assassinate and Death Strike still applies.

1d6(H.Crossbow) + 10(Sharpshooter) + 5(Dex) = 1d6+15

Chance to hit with Precision Strike: 1 - Chance to Miss: 1 - 35% * 35% = 87.75%

Critical: 2d6+15 -> Avg: 22 / Max: 27 -> Times 2 (Death Strike: chance is 65%)

Chance to hit with Precision Strike: 1 - Chance to Miss: 1 - 35% * 35% = 87.75%

Partial Total (multiplied by 87.75% and chance of death strike): Avg: 63.706 / Max: 108

• Action Surge: Attack the surprised target with a poisoned bolt from either hand crossbow using Sharpshooter. Assassinate and Death Strike still applies.

1d6(H.Crossbow) + 10(Sharpshooter) + 5(Dex) = 1d6+15

Critical: 2d6+15 -> Avg: 22 / Max: 27 -> Times 2 (Death Strike: chance is 65%)

Partial Total (multiplied by 87.75% and chance of death strike): Avg: 95.559 / Max: 162

• Radiant Soul: Any attack hits, extra damage from Radiant Soul equal to character level (20)

Chance is 1 - Chances to Miss all 3 = 1 - 35% ^ 6 = 99.82%

Partial Total (multiplied by 99.82%): Avg: 115.523 / Max: 182

• Sneak Attack: Any attack hits, extra damage is 18d6 x 2

Chance is 1 - Chances to Miss all 3 = 1 - 35% ^ 6 = 99.82%

Partial Total (multiplied by 99.82%): Avg: 241.296 / Max: 398

• Purple Worm Poisons

Each attack hits 65% of the time: Half damage on 35% chance save

12d6* 65% * 65% + 12d6 * 65% * 50% * 35% = 22.5225 * 3 = Avg: 67.57 / Max: 72

# Total: Avg: 308.866 — Max: 470

PS: Still working on improvements.

• Comments are not for extended discussion; this conversation has been moved to chat. – mxyzplk Feb 25 '18 at 7:07

# 212.5 points of damage on average (or 644 maximum)

Okay, I'll bite. A Paladin/Fighter (18/2) with Str 20, Cha 20, and Great Weapon fighting style can make 4 attacks and deal 644 points of damage in a single turn:

1. On the prior turn, activate Sacred Weapon (+Cha to hit) and cast Holy Weapon (+2d8 damage to every attack).
2. Attack and extra attack, pumping a divine smite at 4th level into both attacks.
3. Action surge to do all that again, except one divine smite at 3rd level (having run out of 4th-level slots).
4. Use a bonus action to dismiss Holy Weapon and deal 4d8 more points of damage (16.38 average against a Death Knight).

Given the parameters you provided (death knight: AC 20, undead creature), this comes out to 212.5 points of damage on average.

The maximum amount of damage you can do with this combo is 644.

# 189 damage dealt on average (or 301 max).

Without factoring in magic items or artifacts, including Unearthed Arcana, and using a 20th-level straight Fighter build specifically, and not assuming you gain surprise for the attack, the answer appears to be using a 20-Dex Sharpshooter (Martial Archetype) with the fighting style Archery, a longbow, and the feats Sharpshooter (yes, confusing), Silver-Tongued, Elven Accuracy, Lucky, and Martial Adept (Menacing Attack and Pushing Attack). You didn't provide skill values for the opponent, so for simplicity I'll assume any non-combat skill checks are successful.

Your damage for the 'back six' hits is 1d8+5+10+12 which averages to 27.95 each with miss and crit chances factored in, or 167.7; we add to this the initial, assumed hit at 1d8+5+12= ~21.5 for a grand total of 189 damage dealt on average (or 301 max).

Of course, if they're standing on the edge of a long drop, you can use Pushing Attack on the final shot and add 20d6 to that.

• Comments are not for extended discussion; this conversation has been moved to chat. – mxyzplk Feb 25 '18 at 7:07

Average 997.5,Max 1785 Race/Classes/Stats: Very lenient, pick most any race, get to 20. Fighter-2, Cleric-1 (Arcana Domain), Druid-17 (Circle of Twilight from Unearthed Arcana).

Classes explained: As per the rules we grab 2 fighter minimum, which gets us action surge. Cleric Level 1 gets us into the arcana domain, detect magic and Magic missile are always prepared for us. We max out druid levels otherwise, picking the circle of twilight from Unearthed Arcana. Their feature Harvester's Scythe is vital to maximizing our damage. We get a pool of d10's equal to our druid level. Up to half of our maximum may be spent at a time when we roll for damage on a spell, rolling those additional dice and adding them to the damage as bonus necrotic. Yes, we plan on stacking the bonus damage from multiple magic missile darts as with some of the other methods.

Prep round: :)

Go Time- Action: Use 9th level spell slot to cast Magic Missile, creating 3+8 darts, for a total of 11. Roll for damage, add 8d10 to the roll. (17/2=8.5 rounding down for 8 bonus dice maximum) Damage calculation:

D1= 11(1d4+1)+11(8d10), D1avg= 11(2.5+1)+11(8x5.5)= 522.5, D1max= 11(4+1)+11(8x10)= 935

Action Surge: Use 8th level spell slot to cast Magic Missile, creating 3+7 darts, for a total of 10. Roll for damage, add 8d10 to the roll. (17/2=8.5 rounding down for 8 bonus dice maximum) Damage calculation:

D2= 10(1d4+1)+10(8d10), D2avg= 10(2.5+1)+10(8x5.5)= 475, D2max= 10(4+1)+10(8x10)= 850

Totals

Maximum Damage= D1max+D2max = 935+850= 1785

Average Damage= D1avg+D2avg = 522.5+475= 997.5

• Interesting Answer. I suppose Unearthed Arcana wasn't explicitly banned, but I chose not to use it for my answers since it is playtest material. If you are allowed Unearthed Arcana, a higher value can be reached with other techniques. ( @goodguy5 any thoughts?) – David Coffron Apr 29 '18 at 0:57
• I agree. Interesting answer, but UA isn't technically "released" and wouldn't apply. No need to delete or anything. – goodguy5 May 3 '18 at 1:15
• Your pool of d10s is only 17d10 from being a 17th lvl druid. It recharges on a long rest. There's no way you can spend 8 * 21 d10s in a single round, because that's about 10x more than you have in your pool. You spend d10s on a damage roll, not on a whole spell, so they don't get multiplied across multiple damage rolls for a single spell. (I think you could spend them on an AoE, though, and have multiple targets affected by the same damage roll all take the necrotic damage.) – Peter Cordes Jan 5 '19 at 21:10

# Average 443 damage, maximum 1438 damage

Protector Aasimar Paladin 2/Warlock 11/Fighter 3/Ranger 3/Sorcerer 1 (now with less multiclassing (but only slightly))

AS Requirements: Str 20 Cha 20, Wis 13, Dex 13

Required Choices: pact of the blade (warlock pact boon), Tenser's transformation as 11th level Mystic Arcanum (warlock feature), Battle Master (fighter archetype), Disarming Attack (or any other Battle Master talent that adds your superiority die to damage), hunter's mark (ranger spell), Hunter (ranger archetype), colossus slayer (ranger feature), wild magic (sorcerer bloodline), Dual Wielder (Feat), Two-Weapon Fighting (fighting style)

Required Equipment 2 lances, warhorse mount (i realized you can have this because its technically not magical)

Setup: Radiant Burst, Hunter's mark the target and cast Tenser's transformation on yourself. Because Tenser's transformation is technically a wizard spell, roll on the Wild Magic table and get a 93-94, increasing your size to Large.

The Round of Truth: This is surprisingly similar to the other setup. First, make 3 lance attacks (one normal, one TWF, one Tenser's), burning all of your 2nd level spell slots to Divine Smite and all of your warlock slots to Eldritch Smite, and adding all your superiority dice to damage, giving you, per attack:

18 (2d12+5) base + 13 (2d12) Tenser's + 3.5 (1d6) Hunter's Mark + 4.5 (1d8) Colossus Slayer + 27 (6d8) Eldritch Smite + 13.5 (3d8) Divine Smite + 4.5 (superiority die) = 84 damage per attack * 3 attacks + 20 from Radiant Burst = 272 damage, before even Action Surging.

Then you Action Surge, using 3 1st-level slots to Divine Smite (and unfortunately, you're out of slots to Eldritch Smite and superiority dice) but you can still deal:

18 (2d12+5) base + 13 (2d12) Tenser's + 3.5 (1d6) Hunter's Mark + 4.5 (1d8) Colossus Slayer + 9 (2d8) Divine Smite * 3 attacks = another 147 damage.

Add on the 23 damage your horse averages in a turn, the grand total is 272 + 147 + 23 = an average of 443 damage per turn.

Of course I'm not done. I have to calculate what the maximum damage with crits. Here it is in all its glory:

(53+48+12+16+96+48+16)*3+20+(53+48+12+16+36)*3+56 = 1438 maximum damage with crits!

## Average 329, Max 628

Goblin Radiant Sorcerer 17/Fighter 2/Hexblade Warlock 1, with 20 Charisma and the Lucky feat, aided by our plucky Owl familiar, Trilby.

Setup round: use our action to cast Heightened Empowered Delayed Blast Fireball, bonus action to apply Hexblade's Curse to our foe. Trilby takes the Help action, giving my next attack against our foe advantage.

Go time:

Action: Cast Heightened Empowered Meteor Swarm, dealing 40d6+6 damage, with up to 5 rerolls, and a 94% save failure chance (with disadvantage from Heightening and Lucky - 14% chance of using Lucky up)... average 153, max 246

Free: Trigger Delayed Blast Fireball on ourself and our enemy. That's 13d6+6, with up to 5 rerolls, and an 94% chance of them failing the save, gives average 58 max 84. Also hits us, at which point we cast a 6th level Absorb Elements to gain fire resistance.

Action Surge: Cast an 8th level Inflict Wounds on our foe, hitting on a 9+. With Lucky and advantage from Trilby, this is pretty likely (94%). We also have a 27% crit chance, which is pretty great. So we deal 1.21*(10d10 + 6d6) + .94*6 damage, which is average 98, max 278

Fury of the Small: We use this any one time we deal damage to do an extra 20 damage.

Total Average: 329

Total Max: 628

Level 2 Fighter, Level 2 Wizard (Loremaster), Level 6 Cleric (Tempest), Level 10 Druid (Twilight)

• First action, Cast Magic Missile as a 9th level spell. 11 * (1d4+1 force) damage.

• Use Circle of Twilight's Harvest Scythe to add 5d10 to MM damage. 11 * (1d4+1 force + 5d10 necrotic) damage

• Use Loremaster's Spell Secrets to turn necrotic damage to lightning 11 * (1d4+1 force + 5d10 lightning) damage

• Use Tempest's Destructive Wrath to change 5d10 to 50 damage. 11 * (1d4+1 force + 50 lightning) damage

Use Action Surge to repeat above with an 8th level spell, your remaining Harvest Scythe dice, Spell Secrets, and second Destructive Wrath for the following:

10 * (1d4+1 force + 50 lightning) damage.

We can rewrite that as 21 * (1d4+51)

At a minimum you will do 1,092 damage to a single target, at most you will do 1155 damage. Thankfully there isn't much variation, you only have 4 values your damage will be: 1092, 1113, 1134, and 1155.

An average of 1,123.5 isn't bad. If we use the hypothetical death knight with average hp as a target, please line up 5 of them, because this character will only need 4 missiles to down one death knight it can kill all five in a single turn. Since the death knight doesn't have shield as a reaction, resistance to force or lightning damage, there is no question about whether or not it will die from the attacks. If you have doubts on how to apply magic missile damage please see the following:

• Welcome to RPG.SE! Take the tour if you haven't already and see the help center if you need more guidance. Good Luck and Happy Gaming! – Someone_Evil Apr 13 '19 at 19:35
• It might be worth clearly delineating which of the content in your answer (i.e. Lore wizard and Twilight druid) are UA-only content. – V2Blast Apr 13 '19 at 19:52
• Oh its UA all over and some questionable interpretations of rules at that. Is it too much to have faith that someone can google the bold text if they don't know where something came from? Have we that little faith in the general population? – enki Apr 14 '19 at 23:58
• There are actually sixteen different damage results you can get with this build, not only four. You'd roll two different d4s, one for each Magic Missile casting, and since they get multiplied by different numbers of darts, it matters which die is which. Each pair of results on the dice gives a slightly different amount of damage. This doesn't change the average or maximum damage, it just means there are more values in between the min and max you've shown. The full damage is 1050 lightning and 11*1d4 + 10*1d4 + 21 force. – Blckknght Jul 21 '19 at 20:08

## The Character

Camila is a level 20 Goblin {ASs: 10, 20, 12, 10, 10, 18} with these classes...

• Bard (College of Valor) 18 ---------- [for Spells, Battle Magic, the Alert feat, and 3 ASI]
• Fighter 2 ------------------------------ [for Action Surge and the Archery Fighting Style]

...the following Magical Secrets:

• inflict wounds
• conjure woodland beings
• wish

...the following equipment:

• 24 holly berries
• 2 antimatter rifles
• ample energy cells

...and she has participated in downtime Training to gain proficiency with antimatter rifles.

## The Preparation

1. Camila casts wish replicating the effect of hallow with Energy Vulnerability on necrotic damage within 60 feet of the target.
2. Camila uses Action Surge.
3. Camila casts conjure woodland beings with an 8th level spell slot conjuring 24 pixies.
4. The pixies who have lower initiative use Superior Invisibility.

## The Round

1. Camila flanks with one of the pixies and casts inflict wounds with a 7th level spell slot.
2. Camila steps away and attacks with an antimatter rifle using Battle Magic.
3. The pixies take turns picking up, firing, and then dropping the other antimatter rifle (reloading when necessary).

## The Calculation

• All necrotic damage is doubled from hallow's Energy Vulnerability...
• The target never enters the area
• Chance target goes first: 6.125% (using optional rule for ties to avoid GM fiat)
• Chance to save: +10 vs 19 DC = 60%
• totals to 1 - (0.06125*0.6) = 96.325% chance
• Inflict wounds deals...
• 9d10 necrotic damage with a +10 chance to hit (+4 Charisma, +6 Proficiency Bonus) and advantage on the attack roll since it is almost impossible that Camila acts before every pixie.
• Vulnerability * Regular Damage * Regular Hit Chance + Critical Damage * Critical Hit Chance (CHC)
• Expected: 1.9265*[(9*5.5)*(1-45%2-CHC) + (18*5.5)*(1-95%2)] = 88.3575 [Max: 360]
• Camila's antimatter rifle attack deals...
• 6d8 necrotic damage with a +13 chance to hit (+5 Dexterity, +6 Proficiency Bonus, +2 Archery)
• Expected: 1.9265*[(6*4.5)*(65%) + (12*4.5)*(5%)] = 39.01163 [Max: 96]
• Fury of the Small deals...
• 20 necrotic damage, relies on either inflict wounds or the antimatter rifle attack to hit
• Expected: 1.9265*[20*(1-(16%*35%)] = 35.94752 [Max: 40]
• The pixies' antimatter rifle attacks deal...
• 6d8 necrotic damage with a +5 chance to hit (+5 Dexterity) and advantage on any with lower Initiative than Camila due to Unseen Attackers
• Expected for invisible pixies: 1.9265*[(6*4.5)*(1-70%2-CHC) + (12*4.5)*(1-95%2)] = 31.59941 [Max: 96]
• Expected for visible pixies: 1.9265*[(6*4.5)*(25%) + (12*4.5)*(5%)] = 18.20542
• Expected number of invisible pixies is 19.68
• Overall expected damage is 19.68*31.59941 + 4.32*18.20542 = 700.5238 [Max: 2304]

Total expected: 860.07684 [Max: 2800]

Note: This assumes that the target chooses to make the Charisma save against hallow; it doesn't have to. If not, the expected damage increases to 889.7

## Other Summons

Spells like conjure woodland beings don't give the player full control over what creatures specifically get summoned as discussed in this Q&A. If the GM decides to cause a different CR 1/4 or lower creature, the expected damage changes as follows:

• Sprites yield an expected damage of 783.06009 (using Invisibility instead of Superior Invisibility).
• Blink dogs yield an expected damage of 206.62289 (flanking with Bite instead of using the antimatter rifle).
• Boggles yield an expected damage of 692.97184 (using Dimensional Rift to attack without being seen, 11 boggles have to reload instead).

If the GM chooses randomly, you get...

• An expected damage of 635.68291 when all fey at or below CR 1/4 are eligible.
• An expected damage of 616.5866 when only fey at exactly CR 1/4 are eligible.
• An expected damage of 778.70292 if you can guarantee you won't get blink dogs.
• This is a fantastic idea that will almost certainly make your DM say "that was funny, never again." – goodguy5 Nov 7 '19 at 13:44

For my own personal baseline:

Level 20 Fighter - Champion
20 Strength
7 ASIs - +2str, Mounted Combat, Lucky, +1str/+1con, Great Weapon Master, Lucky, Blade Mastery
Greatsword attack - +12 to hit (or +7 with GWM), 2d6ish+5 (or +15 w/GWM)

Things doing: 1. walk horse up to target.
3. Action Surge

5*.64*(2d6+15+.28*2d6) + 3*.78*(2d6+15+.39*2d6)

For an expected 144 in one round. 312 Max

Thought I'd try a stab at this. I know there is a lot of multi-classing, but the goal was to create the biggest expected one turn hit.

# Expected damage is 584.29. Maximum: 1044; here’s how:

## The Character

Goblin (13, 20, 14, 8, 13, 13) + alert feat (Level 11 Spellcaster)

Fighter 5: Battle Master (Action Surge, Multi-attack, Superiority Dice, Dueling, ASI)

Bard 4: College of Whispers (Jack of All Trades, Psychic Blades, ASI)

Sorcerer 4: Divine Soul (For Spells, Quicken/Empowered, 4 Sorcery Points, Favored by the Gods, ASI)

Rogue 3: Assassin (Assassinate, Sneak Attack)

Cleric 2: Grave Domain (Channel Divinity: Path to the Grave)

## The Preparation

Action: Create Bonfire

Movement: Walk in to Bonfire and choose to fail save

Reaction: Absorb Elements: level 6

## The Round

Action Surge: Channel Divinity: Path to the Grave

Action: Multi-attack

## The Calculation

Quickened, Empowered Green Flame Blade w/ Shadow Blade w/ Empowered Absorb Elements, Smite, Psychic Blades, Sneak Attack, Superiority Die, Fury of the Small (Autocrit, Vulnerable, 4 Sorcery Points)

4d8 + 1d8 + 2d6 + 2d6 + 7 psychic + 3d8 + 6d6 Fire + 5d8 Radiant + 20

(26d8 + 20d6 + 27 + (2.8058 + 2.3763 average damage from empowering)) x2

438.3642 Avg (710 max) (59.6116 from GFB)

4d8 + 1d8 + 7 psychic + 5d8 Radiant

(20d8 + 7) x2

97 Avg (167 Max) PerAttack

Chance to 1st Hit (Advantage + Favored by the Gods): 97.15%

Chance to 2nd Hit (Advantage + Favored by the Gods if unused): 95.046%

Chance to 3rd Hit (Advantage + Favored by the Gods if unused): 93.279%

Chance at least 1 hits: 99.94%

Chance at least 2 hit: 99.67%

Chance all 3 hit: 86.13%

Initiative + 13 (dex +5, alert +5, Jack of All Trades +3)

94.75% of going first

(378.7526*0.9994 + 59.6116*0.9715 + 97*0.9967 + 97*0.8613) * 0.9475

### 584.29 Expected (1044 Max)

Caveat: Assumed damage done if second on initiative is 0, thus true value is somewhat higher. Also if enemy is holding shield disarming strike could be used to lower AC on subsequent attacks.

• "Movement: Walk in to Bonfire and choose to fail save" - By RAW, you can't choose to fail a saving throw. – V2Blast Jan 12 '19 at 5:36
• Also, welcome to RPG.SE! Take the tour if you haven't already. – V2Blast Jan 12 '19 at 5:39
• I suppose instead of using Create Bonfire you could use a Firebolt to shoot your own hand. – noisebelt Jul 21 '19 at 16:59

# Expected damage is 709.82. Maximum: 1256; here’s how:

## The Character

Wood Elf (13, 20, 11, 8, 14, 13) + alert, elven accuracy (Level 11 Spellcaster)

Ranger 5: Gloom Stalker (Dread Ambusher, Multi-attack, ASI)

Bard 4: College of Whispers (Jack of All Trades, Psychic Blades, ASI)

Cleric 4: War Domain (Magic Weapon, War Priest, Channel Divinity: Guided Strike, ASI)

Rogue 3: Assassin (Assassinate, Sneak Attack)

Fighter 2: (Action Surge, Great Weapon Fighting)

## The Preparation

Action: Apply Purple Worm Poison

Bonus Action: Magic Weapon: level 6 (C)

## The Round

Action: Attack

Action Surge: Attack

Bonus Action: War Priest bonus attack

## The Calculation

Chance to hit: 98.4375% Chance to miss: 1.5625%

Chance to hit 7 times: 89.562%

6 Times: 9.951%

5 Times: 0.474%

4 Times: 0.013%

With Channel Divinity 99.513% to hit 7 times

65% to fail DC 19 save against poison

2d4 + 8 + 2d6 Slashing + 5d8 Radiant + 2d6 Psychic + (12d6) Poison

2d4 + 8 + 5d8 + (12d6)

2d4 + 1d8 + 8 + 5d8 + (12d6)

2d4 + 8 + 5d8 + (12d6)

2d4 + 8 + 5d8 + (12d6)

2d4 + 1d8 + 8 + 4d8 + (12d6)

2d4 + 8 + 4d8 + (12d6)

If going first:

Total: 28d4 + 70d8 + 8d6 + 56 + (12d6*4.55 fails + 2.45*21 successes)

483 + (4.55*42 + 2.45*21)

722.55 * .99513 chance to hit all 7

Expected 722.016

If going second (no advantage, no crits, 5 hits)

Total: 10d4 + 25d8 + 4d6 + 40 + (60d6)

196.5 + (3.25*42 + 1.75*21)

369.75

Expected: 233.98

Initiative + 15 (dex +5, wis +2, alert +5, Jack of All Trades +3)

97.5% of going first

0.975*722.016 + 0.025*233.98

### 709.82 Expected (1256 Max)

• The double-bladed scimitar is not a finesse weapon and therefore cannot be used with sneak attack. – David Coffron Feb 14 '19 at 14:51
• Also, are your calculations using Channel Divinity more than once? A level 4 cleric can only use it once – David Coffron Feb 14 '19 at 14:54
• I recalculated your damage (using a rapier with dueling instead of a double-bladed scimitar with GWM) and got 474.67 expected. I'm not sure who's right, but you can check my work. I'd love to see how you calculated differently. – David Coffron Feb 14 '19 at 15:43
• You should have 66d8 not 70. – David Coffron Feb 14 '19 at 15:45
• I think I see the problem. Purple worm poison can only apply once. "Injury poison can be applied to weapons, ammunition, trap components, and other objects that deal piercing or slashing damage and remains potent until delivered through a wound or washed off." (emphasis mine). You've applied it multiple times – David Coffron Feb 14 '19 at 15:48

I will start by giving credit where it is due. I can't comment yet (new account) but most of the setup was thought of by David Coffron, I just found a way to improve his damage with Unearthed Arcana's Lore Mastery wizard. Give him all the love please! If I made an error when modifying his approach please let me know.

# Expected damage is 782. Maximum is 1238.

Here's how:

### The Character

Chloe is a level 20 Goblin {ASs: 9, 18, 10, 20, 8, 14} with these classes...

Wizard 17 (Lore Mastery subclass) - [for spells, Alchemical Casting, Alert feat and 3 ASIs]
Fighter 2 - [for Action Surge]
Warlock 1 - [for Hexblade's Curse]

### The Preparation

Chloe casts wish, replicating the effect of hallow with Energy Vulnerability on force damage within 60 feet of the target.

### The Round

Chloe casts magic missile (level 8 spell slot). Chloe uses Alchemical Casting on magic missile (level 1 spell slot)
Chloe uses Action Surge.
Chloe casts magic missile (level 7 spell slot). Chloe uses Alchemical Casting on magic missile (level 1 spell slot)

### The Calculation

All force damage is doubled from hallow's Energy Vulnerability...

• The target never enters the area
• Chance target goes first: 15.125% (using optional rule for ties to avoid DM fiat)
• Chance to save: +10 vs 19 DC = 60%
• totals to 1 - (0.15125*0.6) = 90.925% chance

Each dart from the first magic missile deals...

• 1d4 + 1 + 2d10 (Alchemical Casting) + 6 (Hexblade's Curse) force damage
• 10 darts, doubled 90.925% of the time = 19.90925 multiplier
• Expected: (2.5 + 7 + 11(2*5.5))*19.0925 = 391.39625 [Max: 610]

Each dart from the second magic missile deals...

• 1d4 + 1 + 2d10 (Alchemical Casting) + 6 (Hexblade's Curse) force damage
• 9 darts, doubled 90.925% of the time = 17.18325 multiplier
• Expected: (2.5 + 7 + 11)*17.18325 = 352.256625 [Max: 558]

Fury of the Small deals...

• 1 use per short rest, doubled 90.925% of the time = 1.90925% multiplier
• Expected: 20*1.90925 = 38.185 [Max: 40]

Total expected: 781.837875 [Max: 1238]

• You only roll damage for magic missle once so Alchemical Casting could only add 11 damage since it makes the spell do added damage rather than adding to the damage roll itself, similar to what Fury of the Small does (much less than the benefit from Empowered Evocation); not to mention Lore Mastery is play test material. – David Coffron Mar 14 '18 at 14:51

Assuming you convince your DM to allow some physics, the mach 212 tensors floating disk of stones with 100,635d6 is a good contender. https://www.dandwiki.com/wiki/Pushing_the_Speed_Limit_(5e_Optimized_Character_Build) You might have to remove some damage since you don't allow magic items or boons. But a few tenthousand d6 more or less aren't gonna matter so much. It wouldn't be adventurer league legal, but that wasn't a requirement.

I'm pretty sure any DM is only gonna allow that once. It's fun to think about anyway, so I thought it should be mentioned here.

Expected damage: 446.5

Max damage: 1788

Max with 2 turn set up: 2592

How:

Goblin: Bard 13/sorc 4/fighter 2/ warlock 1

Expected damage:

Turn 1: hex blade curse, contagion (flesh rot for vulnerable all damage) gained through being a bard

Turn 2: bestow curse -> scorching ray (lvl9) -> action surge -> scorching ray (lvl8)

Lvl 9 shoots 10 rays, level 8 shoots 9 so there are 19 rays total and eavh ray each ray does

One ray does 40 extra because of fury of the small and vulnerability

Assuming a 60% hit rate because of their 20ac and us not having magical items

Works out to: 446ish

If you have an extra turn (and access to the spelldriver feat):

Turn 2: bestow curse, hex

Turn 3: scorching ray (9), AS, scorching ray (8), quicken scorching ray (2)

With max dice and crits you get:

((((2d6+1d6+1d8)x2)+6)x2)x22rays +40

Or

2592

(This is assuming pheonix sorcerer from unearthed arcana is NOT allowed, because it allows you to add your cha to each fire damage roll you make... adding another 22x10 damage and making the total 2812)

• Welcome to RPG.SE! Take the tour if you haven't already. Note that you can't quicken a spell and cast another leveled spell as an action on the same turn, due to the bonus-action spellcasting rule. (You can cast two spells as actions using Action Surge, though.) Also, it seems odd to assume that an unofficial (third-party) feat would be allowed in your calculations. – V2Blast Jan 19 '19 at 10:54
• Hello and Welcome! Comtation was erratad. You need at least 3 turns before the flesh rot sets in. – Ruse Jan 19 '19 at 11:14

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# Beginner R: functions that make your life easier

## Beginner R: functions that make your life easier

Let’s get to know my top 10 R’s neat little functions and tricks that make our life easier when manipulating data in R.

### Sequences

Want to make long sequences of numbers or letters but don’t feel like writing them all out into a vector?
R let’s you make a sequence with “:” for numbers. You can also use seq() if you are looking for a regular sequence that is not incremented by one. letters

 [] let’s you make continuous letter sequences, in order or in reverse and starting from where you want in the alphabet. For uppercase letters, you just need to call LETTERS[] instead of letters ☺

> 1:5
[1] 1 2 3 4 5

> seq(1, 10, 2)
[1] 1 3 5 7 9

> letters[1:5]
[1] "a" "b" "c" "d" "e"

> LETTERS[1:5]
[1] "A" "B" "C" "D" "E"

### Paste

This function lets you concatenate strings and variables.
One cool thing about paste(), is that it can be put inside other functions and loops.
For example you want to make a loop, and for each turn of the loop, you want to output a file, named differently (file1, file2, file3). Here is how you go about it:

for (i in 1:10){
write.table(x[i],paste("file", i, ".txt", sep="")
}

What is important to know here is that for paste you need to specify the separator.

> for (i in 1:3) {
print(paste("file", i, ".txt", sep=""))
}
[1] "file1.txt"
[1] "file2.txt"
[1] "file3.txt"

> for (i in 1:3) {
print(paste("file", i, ".txt", sep="_"))
}
[1] "file_1_.txt"
[1] "file_2_.txt"
[1] "file_3_.txt"

### Working directories

Want R to put output files or get output files from a directory specifically?
Look at you, being all organized!
getwd(), for “get working directory”, a function that will print out the current working directory. It’s brother, setwd(), sets the directory to something you specify.

### Substring and stringsplit

Ever had a situation when you want only part of a string? Two useful functions can help you deal with this.
When you have a known mark where you would like to truncate your string, you use strsplit().
When you want everything to be a specific length, you can use substr().
For example:
Gene names that go like this “CPB1_1360” need to be split using the “_character.

strsplit("CPB1_1360", split="_")

String split creates a list of features. You must select the one you want:

> strsplit("CPB1_1360", split="_")[[1]][1]
[1] "CPB1"

> strsplit("CPB1_1360", split="_")[[1]][2]
[1] "1360"

If your separator is a special character, that is a character having a special meaning in a regular expression (like . which means “any” or | which means “or”), you need to use the “\\” or use the option fixed=TRUE.

strsplit("CPB1|1360", split="\\|")
strsplit("CPB1|1360", split="|", fixed=TRUE)

Other example:
Patient barcodes that go like this
“TCGA.E2.A154.01A.11R.A115.07” that you want to shorten to say TCGA-XX-XXXX, you use substr() like this:

substr("TCGA.E2.A154.01A.11R.A115.07", start=0, stop=12)
[1] "TCGA.E2.A154"

Membership of lists
Membership of lists can be tested. When you have two lists, a and b, and want to know which elements in list a are in list b? The syntax is a%in%b. This returns a vector of boolean values (true/false) of the same length of a, that specifies if each value of a is in b. Careful, the reverse needs to be written b%in%a. Here is an example:

> a = c("A","B","C")
> b = c("C","D","R")
> a%in%b
[1] FALSE FALSE TRUE

> b%in%a
[1] TRUE FALSE FALSE

### Selection

Want to select a specific column in a dataframe but don’t feel like counting them to know which index to use? That’s what column names and “$” are for! > a bcr_patient_barcode er_status_by_ihc pr_status_by_ihc her2_status_by_ihc 1 TCGA-AR-A1AR Negative Negative Negative 2 TCGA-BH-A1EO Positive Positive Negative 3 TCGA-BH-A1ES Positive Positive Negative 4 TCGA-BH-A1ET Positive Positive Negative > a$er_status_by_ihc
[1] "Negative" "Positive" "Positive" "Positive"

Another example is selecting features in an object.
For example, you want to output the p-value of a t.test: t.test$p.value. > t.test(e1[,1],e2[,2]) Welch Two Sample t-test data: e1[, 1] and e2[, 2] t = 0.0581, df = 1645.314, p-value = 0.9537 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.02670277 0.02833160 sample estimates: mean of x mean of y 0.5019641 0.5011497 > t.test(e1[,1],e2[,2])$p.value
[1] 0.9537151

To check how the features are called, use names(object) like this:

> names(t.test(e1[,1],e2[,2]))
[1] "statistic" "parameter" "p.value" "conf.int" "estimate" "null.value" "alternative"
[8] "method" "data.name"

### Distributions and random sampling

To get a series of random numbers coming from a normal distribution, call rnorm(), specifying the number of variables, the mean and the standard deviation.

### Help

If you are lost and not sure of what a function does, typing ? in front of the function name opens up the help manual for that function, giving you the details for it.

### Upper case, lower case

If you want to put all letters in a string to upper or lower case, toupper() and tolower() is very handy.

> toupper("RealLy UglY fOrmaT")
[1] "REALLY UGLY FORMAT"

> tolower("MAke Me sMALLeR")
[1] "make me smaller"

### Repetitions and patterns

rep() lets you make custom repetitions. You can specify a pattern and then either specify the number of repeats or specify the total output length with length.out.
Let’s take the most repetitive pop song as an example (repeat 4 times with a high-pitched voice)

> rep(c("I'm like a bird, I'll only fly away", "I don't know where my soul is, I don't know where my home is"),time=4)
[1] "I'm like a bird, I'll only fly away"
[2] "I don't know where my soul is, I don't know where my home is"
[3] "I'm like a bird, I'll only fly away"
[4] "I don't know where my soul is, I don't know where my home is"
[5] "I'm like a bird, I'll only fly away"
[6] "I don't know where my soul is, I don't know where my home is"
[7] "I'm like a bird, I'll only fly away"
[8] "I don't know where my soul is, I don't know where my home is"

By | 2017-04-29T17:06:48+00:00 January 28, 2016|Categories: Bioinformatics, R|Tags: |0 Comments


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# Tag Info

1

If you're willing to expand your definition of money, then yes, and easily: You can get broad money measures (M2, M3 and M4) as a percent of nominal GDP ($\frac{M}{PY} \times 100%$) from the World Bank for several decades for many countries (mileage will vary depending on particular country). Just divide by 100 and invert. By the quantity theory of money, ...

0

According to standard monetary equation: $$MV=PY$$ Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ real output. And where the‘value’ of money is inversely proportional to the price level $P$, we can rearrange the above for $P$ to see how it changes when parameters change and from that we can infer what happens to the value of ...

-2

Both are right. the FED is indeed not printing anything; i assure you no green notes were made in this process. it's all electronic and the accounting checks out of course. the first part of the comment you highlight is a big shortcut, the second which deals with seigniorage is more in tune with reality so let's start with that. The Fed has a around 4T of ...

Top 50 recent answers are included


text:
## A simple Python program¶

We will now write a program -- a series of instructions that Python will execute sequentially.

In [1]:
exam_grade = 98

78

First we assign 98 to exam_grade and 20 to engsci_adjustment. Then we compute (exam_grade - engsci_adjustment), and assign that to exam_grade. Fianlly, we print exam_grade.
exam_grade = exam_grade - engsci_adjustment
is not an equation (i.e., we can't conclude that engsci_adjustment is 0 or anything like that). Rather, it's an instruction: evaluate (exam_grade - engsci_adjustment), and put whatever value you get into the variable exam_grade, forgetting what exam_grade was before.
The result is that exam_grade becomes 78, and then that value gets printed.


text:
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# Find the Third Proportional to 2 2/3 and 4 - Mathematics

Course

#### Question

Find the third proportional to 2 2/3 and 4

#### Solution

Let the third proportional to 2 2/3 and 4 be x

=> 2 2/3 , 4, x are in continued proportion.

=> 2 2/3 : 4 = 4 : x

=> (8/3)/4 = 4/x

=> x = 16 x 3/8 = 6

Is there an error in this question or solution?


text:
# How many times can you pour milk among 3 buckets?

This is a problem from usaco, which I solved, but I don't know how to estimate the number of times you can pour milk among the jugs, my solutions just uses a big enough number.

Here's the problem statement:

Farmer John has three milking buckets of capacity A, B, and C liters. Each of the numbers A, B, and C is an integer from 1 through 20, inclusive. Initially, buckets A and B are empty while bucket C is full of milk. Sometimes, FJ pours milk from one bucket to another until the second bucket is filled or the first bucket is empty. Once begun, a pour must be completed, of course. Being thrifty, no milk may be tossed out.

Write a program to help FJ determine what amounts of milk he can leave in bucket C when he begins with three buckets as above, pours milk among the buckets for a while, and then notes that bucket A is empty.

For example, if the input is 8, 9, 10 then the output is 1, 2, 8, 9, 10.

Another example: if the input is 2, 5, 10 then the output is 5, 6, 7, 8, 9, 10.

My approach is to recursively try all possible move sequences, keeping track of the contents of bucket C each time bucket A is empty. Currently I stop the recursion at the arbitrary depth 200. What is the correct depth to stop the recursion at?

• Please replace your code with pseudocode. If I understand correctly, you want to know whether there is an a priori bound on the length of solutions to this problem. If so, please make it clear. – Yuval Filmus Mar 14 '16 at 10:12
• @YuvalFilmus this is my pseudocode. Where it is unclear what it does I wrote comments, such as TreeSet is a red black tree, I think everything else is pretty clear... I try to recursively find all possible distributions of milk in jugs, but I don't know how to estimate recursion depth of the recurse function. – Pavel Mar 14 '16 at 10:14
• Maybe you shouldn't have taken out the code... There are many ways to solve this, including DP with 3 nested loops. And some other ways. – Pavel Mar 14 '16 at 10:18
• We don't answer programming questions here. If your question requires code then it is out of scope here. Your question was about setting a bound on the depth for your particular algorithm, which can be summarized in one sentence instead of as code. If you're unhappy, try to explain your question better. – Yuval Filmus Mar 14 '16 at 10:20

Since the total amount of water is C liters (C being the capacity of bucket C), the number of possible states is at most the number of non-negative integer solutions of $x+y+z = C$, which is $\binom{C+2}{2}$. Any sequence of moves longer than that must repeat a state. Therefore it is enough to recurse up to level $\binom{C+2}{2}$.
• That's a standard result in combinatorics. More generally, the number of solutions of $x_1+\cdots+x_n=C$ is $\binom{C+n-1}{n-1}$. – Yuval Filmus Mar 14 '16 at 10:59