### Pyramids

What are the missing numbers in the pyramids?

### Paving the Way

A man paved a square courtyard and then decided that it was too small. He took up the tiles, bought 100 more and used them to pave another square courtyard. How many tiles did he use altogether?

### Chess

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

# Sum and Differences

##### Age 11 to 14 Short Challenge Level:
Using algebra
Suppose the middle number is $n$. Then the other numbers are $n+12$ and $n-2$.

The three numbers add up to $100$, so \begin{align}n+(n+12)+(n-2)&=100\\ \Rightarrow n+n+12+n-2&=100\\ \Rightarrow 3n+10&=100\\ \Rightarrow 3n&=90\\ \Rightarrow n&=30\end{align} So $n=30$, $n+12=42$ and $n-2=28$.

Choosing the middle number
If the middle number was $50$, then the other two would be $50 + 12 = 62$ and $50 -2 = 48.$ $50 + 62 + 48 = 160.$ That is much too big!

If the middle number was $40$, then the other two would be $52$ and $38.$ $40 + 52 + 38 = 130.$ That is still too big!

If the middle number was $30$, then the other two would be $42$ and $28.$ $30 + 42 + 28 = 100.$

Using numbers which add up to $100$
If we start by looking at numbers which are close together, $33 + 33 + 34 = 100.$

Then leaving the middle number as $33$, the smallest number would need to be $31$ and the largest should be $33 + 12 = 45$, which is the same as $34 + 11.$ So $31, 33, 45$ have the correct 'gaps'.

$33$ has gone down by $2$ to make $31$ and $34$ has gone up by $11$ to make $45$, so the sum has gone down by $2$ and up by $11$, which is the same as going up by $9$ (to $109$).

To reverse the increase of $9$, we can subtract $3$ from each of the numbers, leaving us with $28, 30, 42.$

You can find more short problems, arranged by curriculum topic, in our short problems collection.