Card game-related numbers

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This page contains card game-related numbers. Numbers related to the tile-based games Mahjong and Rummikub are also included.

List of card game-related numbers

• A standard 52-card deck without jokers contains 52! possible ways of being organized, or over 80 unvigintillion.
• There are 104 playing cards in a typical rummy deck without jokers, and 106 tiles with 2 joker tiles.
  • There are \(106! \approx 1.14 \times 10^{170}\) ways to arrange said 106 tiles.
• The highest possible game value (Grand ouvert with all four jacks) in the German card game of Skat is 264.
• A common variant of the German card game of Schafkopf uses 24 cards, of which six cards will be given to each of the four players, resulting in ₂₄C₆ = 134,596 hands for a player.
• There are 9! = 362,880 possible ways to arrange a sequence of moves that fills a \(3\times 3\) tic tac toe board. According to half-real site, 255,168 possible games of tic tac toe if only counting valid games. (where rotations and reflections of other games are counted as different games). Of these, 131,184 are won by the first player, 77,904 are won by the second player, and 46,080 are drawn.
• In the card game of poker, a poker hand can be any combination of 5 cards of a standard 52-card deck. Therefore, the number of possible combinations is equal to ₅₂C₅ = 2,598,960.
  • It is also equal to the number of possible combinations in a 5/52 lottery available in some U.S. states, such as Arizona, Indiana and Kentucky, where you must pick 5 cards out of 52, or they are generated by a terminal.
  • Its prime factorization is 2⁴ × 3 × 5 × 7² × 13 × 17.
• The German card game of Schafkopf uses 32 cards, of which eight cards will be given to each of the four players, resulting in ₃₂C₈ = 10,518,300 hands for a player.
• The German card game of Skat uses 32 cards, of which 10 cards will be given to each of the three players, and two cards remain on the table, resulting in ₃₂C₁₀ = 64,512,240 hands for a player.
• The German card game of Doppelkopf uses 48 cards (two copies of 24 types), of which 12 cards will be given to each of the four players, resulting in 287,134,346 hands.
• The Chinese tile game of Mahjong uses 136 tiles (four copies of 34 types). There are 98,521,596,000 possible combinations for regular hands of 13 tiles, and 326,520,504,500 possible combinations for regular hands of 14 tiles for the dealer.
• The card game of contract bridge uses 52 cards, of which 13 cards will be given to each of the four players, resulting in ₅₂C₁₃ = 635,013,559,600 hands for a player.
• A common variant of the German card game of Schafkopf uses 24 cards, of which six cards will be given to each of the four players, resulting in 24!/(6!)⁴ = 2,308,743,493,056 card distributions.
• The German card game of Skat uses 32 cards, of which 10 cards will be given to each of the three players, and two cards remain on the table, resulting in 32!/(10!)³/2 = 2,753,294,408,504,640 card distributions.
• The German card game of Schafkopf uses 32 cards, of which eight cards will be given to each of the four players, resulting in 32!/(8!)⁴ = 99,561,092,450,391,000 card distributions.
• The card game of contract bridge uses 52 cards, of which 13 cards will be given to each of the four players, resulting in \(52!/(13!)^4\) = 53,644,737,765,488,792,839,237,440,000 card distributions.
• In Mahjong, which has 136 tiles consisting of 34 types with 4 tiles each, the number of possible combinations of tiles in the wall is \(136!/(4!)^{34}\). Full decimal expansion is
  432698391742847263810138305033742131882589880014027980942911799590972036509850338890918625906160332246611492154251485203537706583203911528927630102575646500000000000000000000000000000000

Uno-related numbers

Uno is an American shedding-type card game that is played with a specially printed deck. The game's general principles put it into the Crazy Eights family of card games, and it is similar to the traditional European game Mau-Mau.

• 1: When you have only one card, you must speak Uno.
• 2: There is a card that makes the opponent draw two cards ("+2" or Draw 2).
• 4: There is a card that makes the opponent draw four cards ("+4" or Wild Draw 4).
• 7: Number of cards in initial hand of each player.
• 54: In Classic Uno, there are exactly 54 different cards, and 108 cards in total.
  • 10 numeric cards (0-9) with 4 different colors for 0 and = 40. 1 copy for 0 and 2 copies for 1-9, summing up to 76.
  • 1 "+2" card with 4 different colors = 4. 2 copies = 8.
  • 1 skip card with 4 different colors = 4. 2 copies = 8.
  • 1 reverse card with 4 different colors = 4. 2 copies = 8.
  • 1 Wild card = 1. 4 copies.
  • 1 "+4" or Wild Draw 4 = 1. 4 copies.
• 58: In Uno Rabbids, there are exactly 58 different cards
  • Same as Classic Uno = 54.
  • 1 Comin' Through card = 1.
  • Explosive Results card = 1.
  • Hurry Up! card = 1.
  • Wild Blue Yonder card = 1.
• 349,426,862: Number of possible first hands in Classic Uno.

Approximations of these numbers

For 134,596:

Notation	Lower bound	Upper bound
Scientific notation	\(1.34596 \times 10^5\)
Arrow notation	\(51\uparrow3\)	\(367↑2\)
Steinhaus-Moser Notation	6[3]	7[3]
Copy notation	1[6]	2[6]
Taro's multivariable Ackermann function	A(3,14)	A(3,15)
Pound-Star Notation	#*(36)*3	#*(37)*3
BEAF	{52,3}	{53,3}
Hyper-E notation	E5	E[53]3
Hyperfactorial array notation	8!	4!1
Fast-growing hierarchy	\(f_2(13)\)	\(f_2(14)\)
Hardy hierarchy	\(H_{\omega^2}(13)\)	\(H_{\omega^2}(14)\)
Slow-growing hierarchy	\(g_{\omega^4\times5+\omega\times4}(13)\)

For 53,644,737,765,488,792,839,237,440,000:

Notation	Lower bound	Upper bound
Scientific notation	\(5.364\times10^{28}\)	\(5.365\times10^{28}\)
Arrow notation	\(12\,712\uparrow7\)	\(12\,713\uparrow7\)
Steinhaus-Moser Notation	21[3]	22[3]
Copy notation	4[29]	5[29]
Chained arrow notation	\(12\,712\rightarrow7\)	\(12\,713\rightarrow7\)
Taro's multivariable Ackermann function	A(3,92)	A(3,93)
Pound-Star Notation	#*(8,4,6)*8	#*(8,4,6)*9
PlantStar's Debut Notation	[17]	[18]
BEAF & Bird's array notation	{12712,7}	{12713,7}
Hyper-E notation	5E28	6E28
Bashicu matrix system	(0)(0)(0)[2964]	(0)(0)(0)[2965]
Hyperfactorial array notation	27!	28!
Strong array notation	s(12712,7)	s(12713,7)
Fast-growing hierarchy	\(f_2(88)\)	\(f_2(89)\)
Hardy hierarchy	\(H_{\omega^2}(88)\)	\(H_{\omega^2}(89)\)
Slow-growing hierarchy	\(g_{\omega^{7}}(12\,712)\)	\(g_{\omega^{7}}(12\,713)\)