# concentration (card game)

• Rules can be changed here too: it can be agreed before the game starts that matching pairs be any two cards of the same rank, a color-match being unnecessary, or that the
match must be both rank and card suit.

• Less obviously, if the card does not match any known card, one of the n known cards should still be chosen to minimize the information provided to other players.

• Solitaire Concentration may be played solo either as a leisurely exercise, or with the following scoring method: play as normal, but keep track of the number of non-matching
pairs turned over (this may be done using poker chips, pennies or by making marks on a sheet of paper).

• Before any turn in the game, there are t cards still in play, and n cards still in play but of known value.

• Rules Any deck of playing cards may be used, although there are also commercial sets of cards with images.

• [4] Strategy Over the course of the game, it becomes known where certain cards are located, and so upon turning up one card, players with good memory will be able to remember
where they have already seen its pair.

• • One Flip: Players who make a successful pair win these cards but do not go again until their next turn.

• • Zebra: Pairs may only be formed by cards of the same rank, but opposite in color (so 7 of diamonds would match with a 7 of clubs or spades, but not a 7 of hearts) • Two
Decks: For a much longer game, shuffle together two 52-card decks and lay them out in 8 rows of 13 cards (9 rows of 12 cards if using jokers).

• With perfect memorization and using an optimal strategy, the expected number of moves needed for a game with cards converges to , with .

• Variations Many of these may be played in combination with one another: • Any Color: A version especially good for young children where matching pairs need only be of the
same rank, not the same color.

• Concentration is a round game in which all of the cards are laid face down on a surface and two cards are flipped face up over each turn.

• After every turn roles are exchanged, in this case: player two flips a card and player one answers.

• The rules given here are for a standard deck of 52 cards, which are normally laid face down in four rows of 13 cards each.

• Pairs must be identical (same rank and same suit, so the 10 of clubs would have to match the other 10 of clubs).

• Pairs must be identical (same rank and same suit, so the 10 of clubs would have to match the other 10 of clubs).

• In this case, her opponent must say which cards have been turned up, turn the cards back down if she doesn’t get a pair and take the cards out for her if she does get a pair.

• The mathematics follow: If a remaining unknown card is chosen randomly, there is a 1/(t−1−n) chance of getting a match, but also a n/(t−1−n) chance of providing opponents
with the information needed to make a match.

Works Cited

[‘1. “Pelmanism” in 50 Card Games: 50 Popular Card Games for Hours of Fun. Igloo Books. 2018. p. 23. ISBN 9781784409852.
2. ^ Concentration Solitaire Rules, Solitaire Central. Accessed 14 October 2020
3. ^ Daniel J. Velleman; Gregory S. Warrington
(2013). “What to Expect in a Game of Memory”. The American Mathematical Monthly. 120 (9): 787. doi:10.4169/amer.math.monthly.120.09.787. S2CID 207521103.
4. ^ Foerster, K.-T.; Wattenhofer, R. (2013). The Solitaire Memory Game (Technical report).
ETH Zurich.
5. ^ Chazottes, Jean-René; Collet, Pierre; Redig, Frank (2021-06-25). “Evolution of Concentration Under Lattice Spin-Flip Dynamics”. Journal of Statistical Physics. 184 (1). arXiv:2103.08264. doi:10.1007/s10955-021-02796-0. ISSN 0022-4715.

Photo credit: https://www.flickr.com/photos/kjunstorm/1793211631/’]