Problem
COM-B1-M08-P009 Numbers from \(1\) to \(20\)
#9
★★☆☆☆ Level 2 of 5
Players alternately choose one previously unchosen number from \(1,2,\ldots,20\). After all numbers are chosen, they compare their sums. Prove that the second player can guarantee a draw.
Pair the numbers by sum \(21\).
The second player pairs the numbers as \((1,20),(2,19),\ldots,(10,11)\). Whenever the first player takes a number from a pair, the second takes the other number in that pair. Each pair sums to \(21\), and each player gets exactly one number from each pair. Therefore the final sums are equal.
Pure pairing strategy without a board.