The X+Y Files

Issue 4

• A coin puzzle
• Numbers with names - perfect, abundant or deficient
• Number puzzles
• Multiplication puzzle
• Colour maze
• Mathematical crossword
• Old favourites - how many?

How Many?

How many squares are there in this diagram? The quick answer is 16, but this does not include squares of all sizes. The total number of squares is 1 + 4 + 9 + 16 = 30. It follows that in an n × n square grid there are 1 + 2² + 3² + ... n². [There is a formula for this sum: 1/6n(n + 1)(2n + 1)]

In this triangular grid the total number of triangles is 16 + 7 + 3 + 1 = 27 The problem is more complicated because the triangles can appear two ways up –  and . The number of s = 10 + 6 + 3 + 1 = 20 The number of s = 6 + 1 = 7 Both these totals are sums of triangular numbers. There is a formula for the sum of the first n triangular numbers, and hence for the number of s.

1 + 3 + 6 + 10 + … + ¹/₂n(n + 1) = ¹/₆n(n + 1)(n + 2) 

However, the number of s depends on whether n is odd or even.