Puzzle One

The board shown below has 32 cells, one of which is labelled S (Start) and another F (Finish). The shortest path starting at S and finishing at F involves exactly nine other cells and ten moves, where each move goes from cell to cell ‘horizontally’ or ‘vertically’ across an edge.

	From S, there are two starting paths:
	We can represent the number of paths to a particular square like this:
There is only one way to travel along the edges and we can build up our diagram like this, noticing that each square not on an edge can only be reached from the left or from below.
How many possible paths are there from S to F by using the instructions above?

Solution:

Since F is 5 steps above and 5 steps to the right of S, each of the 10 moves can only be upwards or to the right and so there are a total of 52 paths from to S to F.         

Puzzle Two

There are three circles with centre points A, B and C. Each circle touches the other two circles, and each centre lies outside the other two circles. The sides of the triangle, drawn connecting the 3 points, have lengths 13 cm, 16 cm and 20 cm. What are the radii of each of the three circles?

Solution:

Given: a + b = 13, b + c = 16 and a + c = 20

Adding these equations together, we obtain: 2a + 2b + 2c = 49

Dividing both sides by 2, we get: a + b + c = 24.5,

then, c = 24.5 – (a + b) = 24.5 – 13 = 11.5 (Radius for circle C)

And, b = 16 – c = 16 – 11.5 = 4.5 (Radius for circle B)

And, a = 13 – b = 13 – 4.5 = 8.5 (Radius for circle A)

Puzzle Three

A large square is split into four congruent squares, two of which are shaded. The other two squares have smaller shaded squares drawn in them whose vertices are the midpoints of the sides of the unshaded squares. What percentage of the diagram is shaded?

Solution:

If you divide the large outside square as shown with red lines, there are 4×4=16 red line divided squares. The shaded square are 12 of the 16 (2×2 and 2×4) total squares. Therefore, the fraction of shaded to total squares is 12/16 = ¾ or 75%.