### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

### Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

### The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

# Mind Your Ps and Qs

##### Stage: 5 Short Challenge Level:

Here are 16 propositions involving a real number $x$:

|                          |                        |             |                         |
|--------------------------|------------------------|-------------|-------------------------|
| $x\int^x_0 ydy < 0$      | $x> 1$                 | $0< x< 1$  | $x^2+4x+4 =0$           |
| $x=0$                   | $\cos(x/2)> \sin(x/2)$ | $x> 2$      | $x=1$                   |
| $2\int^{x^2}_0ydy> x^2$ | $x< 0$                | $x^2+x-2=0$ | $x=-2$                 |
| $x^3> 1$                 | $|x|> 1$               | $x> 4$      | $\int^x_0 \cos y dy =0$ |

[Note: the trig functions are measured in radians]

By choosing $p$ and $q$ from this list, how many correct mathematical statements of the form $p\Rightarrow q$ or $p\Leftrightarrow q$ can you make?

It is possible to rearrange the statements into four statements $p\Rightarrow q$ and four statements $p\Leftrightarrow q$. Can you work out how to do this?

NOTES AND BACKGROUND

Logical thinking is at the heart of higher mathematics: In order to construct clear, correct arguments in ever more complicated situations mathematicians rely on clarity of language and logic. Logic is also at the heart of computer programming and circuitry. To find out more, look at the ideas surrounding the Adding Machine problem and related set of activities.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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