### 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.

### Chocolate Maths

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it works?

# Pick

##### Stage: 3 Challenge Level:

Investigate polygons, like those in the diagrams, with all the vertices on the lattice points of a grid. For each polygon, work out the area A, count the number B of grid points on the boundary, and count the number I of grid points in the interior of the polygon. Can you find a formula connecting A, B and I? Display your results in a table, for example:

A B I
16 8 13
64.5 13 59

The following method may help you to find a formula if you do not spot the pattern. First divide your polygon into triangles each of which has an area of one half a square unit. Next consider the total sum of all the angles in all the triangles in two different ways. If you assume that any polygon can be split into triangles in this way, then this method gives a proof of a general formula connecting A, B and I.