Original Card Games by David Parlett
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Players 2   Cards 52   Type Arithmetical

The grand old Duke of York,
He had ten thousand men -
He marched them up to the top of the hill
And he marched them down again

Use 52 cards counting as follows: Q=0, A=1, 2-10=face value, Jack=11, K=12.
Deal five cards each, face down and one at a time, then two more face down to the table, and stack the rest face down.
To play the last card in each of five rounds.
Each in turn, starting with non-dealer, takes one of the two table cards, adds it to their hand, and discards any one card face down in its place. This may enable you to improve your hand when you have played often enough to be able to make that judgement.

Non-dealer then plays a card to the table and announces its value. Thereafter each in turn adds a card to those played and announces their cumulative total. If you can add another card without exceeding 31 you must. If not, the next card you play must be deducted from the current total, as must each subsequent card played. If you can play without going below zero you must, otherwise you lose. Whoever plays the last card wins that round.
Score 1 for reaching the top of the hill (making the highest total), or 2 for making 30, 3 for 31. Score 1 for being the first back down (playing the last card), or 2 for making 1, or 3 for 0.
After each round set aside all ten cards just played in that deal, leaving the two discards in place ready for exchange in the next round. The next in turn to deal does so and play proceeds as before.
The winner is the player with the highest score after five rounds.
Optional extras
(1)You may like to play each hand twice with the same cards but with the other player leading. In this case a game is ten deals instead of five.

(2) You may prefer to skip Queen=0 and count all cards at their face value from Ace=1 to J=11, Queen=12, King=13. But Queen=0 is preferable because it sets the average value of all cards to 6 and therefore requires an average play of ten cards to get up to 30 and back down again.
Notes on play
A good hand is one with a wide range of numerals. In exchanging your first card before play begins it's therefore a good idea to discard a numeral of which you have two. If you're dealt mostly high cards, the exchange may enable you to pick up a low one, otherwise you're likely to run out of cards before you can get down to the 'bottom of the hill'. The converse also applies.

Queens can be devastating, as you will soon discover! That might be a good reason for preferring Q=11, J=12, K=13.

It is tempting to play all your high cards first so as to hold low ones back for the downhill journey, but a bit of experimentation will show that this may not be the best policy as different results arise from playing cards in a different order. Since each of you can play your own cards in 120 different ways, any given deal allows up to 14,400 different games. (Fewer if some numerals are duplicated, more if you follow the suit-matching variation.)