### 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?

# Nine, Ten and One

##### Age 11 to 14 Short Challenge Level:
t = 5

We are given that 2n + e = 8 and t + e + n =10. Subtracting the first equation from the second gives t - n = 2.

As n cannot equal 1, the minimum value of t is 4 but this gives n = 2, e =4 which is not allowed.

If t = 5 then n = 3 and e = 2, which is allowed.

If t > 5 then n >3 and e is not a positive whole number, so 5 is the only possible value of t.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.