### Pyramids

What are the missing numbers in the pyramids?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

# Kangaroo Subtraction

##### Stage: 3 Short Challenge Level:

The difference between '$KAN$' and '$GAR$' is less than $100$ and since $K\neq G$ we must have $K=G+1$.

Next we must have $N < R$ else the difference between '$KAN$' and '$GAR$' would be at least $100$.

Let $R=N+x$ where $1< x< 9$. Then $OO=100-x$ and hence $O=9$ and $R=N+1$. Also we must have $K\leq 8$

We want the largest value for $KAN$ so we try $K=8$. This forces $G=7$, hence must have $A\leq 6$. Set $A=6$, this forces $R\leq 5$ and hence $N\leq 4$ since $R=N+1$.

So $864$ is the largest possible value for $KAN$, and we have $864-765=99$.

This problem is taken from the UKMT Mathematical Challenges.