From reader Derek Christie:
Each player in a game of cribbage has a hand of four cards. A single further card is turned up and serves as the fifth card in every player’s hand. Part of the game involves scoring your hand. You get points for any combination of cards that adds to 15, like 9 4 2; for two or more of any rank, like 3 3; or for any run of three or more, like Ace 2 3. The Jack, Queen, and King each score 10. Show that if a 5 has been turned up, every player must score some points.

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Try and make a hand that doesn’t score.
The turnup is a 5. This means you must choose 4 different additional cards from the group 1 2 3 4 6 7 8 9, and to avoid scoring, the resulting hand mustn’t include any pairs or runs or any combination of cards that adds to 10.
Since you can’t have two cards adding to 10, you’ll need to choose one card from each of the pairs 9 1, 8 2, 7 3, and 6 4. Try 6 from 6 4. Now you have 5 6. You can’t have 7 (which would produce a run), so you must take the 3, giving 5 6 3. But this is a dead end: Now you must avoid both the 1 and the 9, since each of these produces a 15. So you can’t get anywhere with the 6 from the 6 4 pair.
Try the other card from that pair, the 4. Now you have 4 5. You can’t have 3 (producing a run), so you must take the 7, giving 4 5 7. Now you can’t take 8 (since this would produce 15), so you must take the 2, giving 2 4 5 7. And now both the 1 and the 9 again give 15, another dead end.
With both the 6 and the 4 excluded, there’s no way to assemble a hand of four cards that can remain scoreless with a turnup card of 5. So every hand must score something.
(Thanks, Derek.)
