### Man Food

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?

### Sam Again

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

### Pick

Investigate polygons with all the vertices on the lattice points of a grid. For each polygon, work out the area A, the number B of points on the boundary and the number of points (I) inside the polygon. Can you find a formula connecting A, B and I?

# Carry Over

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

$UK$ means $10U + K$ and $SMC$ means $100S + 10M + C$, so we have $$10U+K+4=100S+10M+C$$ The left hand side is at most $$10 \times 9 + 8 + 4 = 102$$ so $$100S+10M+C \leq 102$$ Therefore $S \leq 1$, so $S=1$ (since it can't be zero). So $$10M+C \leq2$$ So $M=0$

$M$ has the lowest value.

This problem is taken from the UKMT Mathematical Challenges.