### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Plum Tree

Label this plum tree graph to make it totally magic!

### Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

# Sums of Pairs

##### Stage: 3 and 4 Challenge Level:

Here's some sound and efficient algebra from Conor from Queen Elizabeth's Hospital :

We have three numbers, $x, y$ and $z$.

$x + y = 11 \quad (1)$
$y + z = 17 \quad (2)$
$z + x = 22 \quad (3)$

Adding $(1)$ and $(2)$ gives :

$x + 2y + z = 28$

Putting that another way :

$(x + z) + 2y = 28 \quad (4)$

Substituting $(3)$ into $(4)$ gives :

$22 + 2y = 28$

So $y = 3$.

Substituting $y = 3$ into $(1)$ gives $x = 8$ and substituting it into $(2)$ gives $z = 14$.

But Eden has had an excellent approach too :

$11 + 17 + 22$ uses every number twice and makes a total of $50$

So the total of the three numbers that we need to find is $25$.

If two of them make $11$ together the one not used must have been $14$.

Two together made $17$, the one not used this time must have been $8$.

And finally, a pair have a sum of $22$, so the other number is $3$.

The three numbers are $3, 8,$ and $14$

And Mark used some deductive reasoning combined with an exhaustive approach:

The smallest number must be between $1$ and $5$ to make a smallest sum of $11$.
If it is $1$ then to make $11$ the second number is $10$.
If the third number is $x$
Then $x+1= 17$
and $x+10 = 22$
This does not work.

I tried the smallest number as $2$, then the second number is $9$
Then $x+2= 17$ and $x+9 = 22$
This also does not work but the two values of $x$ are closer than last time so I think $3$ will work.

Trying the first number as $3$ and the second as $8$
Then $x+3= 17$ and $x+8 = 22$
This works.

The three numbers are $3, 8$ and $14$.

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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