Earlier today I set you the following two puzzles, inspired by a money-saving trick devised by Carl Friedrich Gauss. (Click to the original for the explanation of the trick, and what it has got to do with Gauss).
1. The double bill.
Which of the two additions below (the sum of the nine numbers on the left, and the sum of the nine on the right) adds to the highest number?
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Solution: The sums are the same. I love this puzzle since the answer seems so unlikely until you start to mouth the calculation. The leftmost column on each sum contains either one 9, or nine 1s, The next column contains two 8s, or eight 2s. Thanks to the commutativity of multiplication, the totals for each column are the same, and thus the totals of both additions are the same.
2. The 2D envelope challenge.
Sum all the numbers in the grid below.
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Solution: 1,000
Here’s one way of doing it. Pair the numbers diagonally from top left to bottom right. (1 + 19) = 20, (2 + 18) = 20, and so on. There are (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45 of these pairs, which makes 45 x 20 = 900. Add the remaining ten 10s, and you get 1000.
I hope you enjoyed these puzzles, I’ll be back in two weeks.
I’m the author of several books of maths and puzzles and also of the kids’ book series Football School. The latest in the series, Terrific Teams, is out this week. It contains stories about 50 amazing teams from Bungay Town to Brazil, and from Lewes to Liverpool.
If you are between 7-12 years old and you like sport and writing, or if you know anyone who is, why not enter the Football School/The Guardian Young Sportswriter of the Year competition. Prizes include your story published by the Guardian. For more information please click here.
