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A pair that fits

BrainteasersEasy

From the whole numbers $1$ to $20$ you pick distinct numbers. What is the smallest number of picks that guarantees that $2$ of the picked numbers add up to exactly $21$ ?

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Hint

Split the numbers into boxes so that a sum of $21$ means $2$ picks in one box.

Solution

The numbers split into $10$ pairs that sum to $21$ : $1 + 20$ , $2 + 19$ , …, $10 + 11$ .
Picking one number from each pair gives $10$ numbers with no $2$ summing to $21$ , for example $1$ to $10$ .
With $11$ picks, $2$ of them fall in the same pair, so the answer is $11$ .

Answer: $11$

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