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?
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 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?
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
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
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
180T = 360I + 360 + 180(B - 4)
Dividing this by 360 gives the formula for the area of the
A = T/2 = I + 1 + (B - 4)/2
A = I + B/2 - 1