Twenty people may stand in any order. Even so, two apparently different counts always end in a perfect tie.
Twenty in a Row
Riddle statement
Ten boys and ten girls stand in a line, in any order.
- Each boy counts how many people are to his right.
- Each girl counts how many people are to her left.
Add together all the numbers obtained by the boys, and separately add together all the numbers obtained by the girls.
Which of the two totals will be larger?
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Solution
Consider the contribution made by each pair of people.
Two boys. The boy on the left counts the boy on his right. Each pair of boys contributes 1 to the boys' total. With 10 boys, there are 10 × 9 ÷ 2 = 45 such pairs.
Two girls. The girl on the right counts the girl on her left. Each pair of girls contributes 1 to the girls' total. There are also 45 such pairs.
One boy and one girl. If the boy is on the left, they both count each other, so the pair contributes 1 to each total. If the girl is on the left, neither counts the other, so the pair contributes 0 to both totals.
The boy-boy pairs contribute 45 to the boys' total, and the girl-girl pairs contribute 45 to the girls' total. Every mixed pair contributes equally to both, so the two totals must always be equal.