The addition chart

Your book, Maths Mela (Chapter 10), has a big addition chart. The numbers 0 to 12 run down the left side (the rows) and across the top (the columns). In each box we write the row number plus the column number. So the box in row 7 and column 5 holds 7 + 5 = 12.

How many times does 9 appear in the chart? Each 9 comes from a way of making 9: 0 + 9, 1 + 8, 2 + 7, 3 + 6, 4 + 5, 5 + 4, 6 + 3, 7 + 2, 8 + 1 and 9 + 0. That is 10 ways, so 9 appears 10 times. In the same way, 5 appears 6 times, from 0 + 5 up to 5 + 0.

Is any row made of only even numbers, or only odd numbers? No. Along a row the numbers go up by 1 each time: 3, 4, 5, 6 … So they always go even, odd, even, odd. The columns do the same.

The book puts a small red window around four boxes: 10 and 11 on top, 11 and 12 below. Add along each diagonal: 10 + 12 = 22 and 11 + 11 = 22. The two diagonals have the same sum! Move the window anywhere, and it still works: the window 14, 15 on top and 15, 16 below gives 14 + 16 = 30 and 15 + 15 = 30.