### Human 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

##### Age 11 to 14 Challenge Level:

We can find the area A of any polygon which has its vertices on the lattice points of a rectangular grid, in terms of the number B of lattice points on the boundary and the number I of lattice points inside the polygon.

This solution was sent by Ling Xiang Ning, Allan from Raffles Institution, Singapore.

First, I divided the polygon into triangles, each with an area of half a square unit. If the number of triangles is T then the area of the polygon is T/2. All the grid points inside and on the boundary of the polygon become vertices of the triangles.

The total angle measurement in each triangle is 180 degrees so the total of the angles in all the triangles is given by 180T. This is made up of the total of the angles at B grid points on the boundary plus the total of the angles at I grid points inside the polygon.

At each grid point inside the polygon the angles of the triangles meeting at that point add up to 360 degrees. There are 4 points at the vertices of the polygon where the angles in the triangles meeting at those 4 points add up to 360 degrees. There are (B - 4) grid points on the boundary of the polygon where the angles in the triangles meeting at each point add up to 180 degrees.

180T = 360I + 360 + 180(B - 4)

Dividing this by 360 gives the formula for the area of the polygon:

A = T/2 = I + 1 + (B - 4)/2
A = I + B/2 - 1