For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12?

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Pai gow

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Pai gow

Chinese

牌九

Literal meaning

card nine

Pai gow (/paɪ ˈɡaʊ/ py GOW; Chinese: 牌九; Jyutping: paai⁴ gau² [pʰaːi˩.kɐu˧˥]) is a Chinese gambling game, played with a set of 32 Chinese dominoes. It is played in major casinos in China (including Macau); the United States (including Boston, Massachusetts; Las Vegas, Nevada; Reno, Nevada; Connecticut; Atlantic City, New Jersey; Pennsylvania; Mississippi; and cardrooms in California); Canada (including Edmonton, Alberta and Calgary, Alberta); Australia; and New Zealand.

The name pai gow is sometimes used to refer to a card game called pai gow poker (or "double-hand poker"), which is loosely based on pai gow. The act of playing pai gow is also colloquially known as "eating dog meat".^[1]

History[edit]

Pai Gow is the first documented form of dominoes, originating in China before or during the Song Dynasty.^[2] It is also the ancestor of modern, western dominoes. The name literally means "make nine"^[3] after the normal maximum hand, and the original game was modeled after both a Chinese creation myth, and military organization in China at that time (ranks one through nine).

Rules[edit]

Starting[edit]

Tiles are shuffled on the table and are arranged into eight face-down stacks of four tiles each^[4]^: 203 in an assembly known as the woodpile. Individual stacks or tiles may then be moved in specific ways to rearrange the woodpile, after which the players place their bets.

Next, each player (including the dealer) is given one stack of tiles and must use them to form two hands of two tiles each. The hand with the lower value is called the front hand, and the hand with the higher value is called the rear hand. If a player's front hand beats the dealer's front hand, and the player's rear hand beats the dealer's rear hand, then that player wins the bet and is paid off at 1:1 odds (even money). If a player's front and rear hands both lose to the dealer's respective hands, the player loses the bet. If one hand wins and the other loses, the player is said to push, and gets back only the money he or she bet.^[5] Generally seven players will play, and each player's hands are compared only against the dealer's hands; comparisons are always front-front and rear-rear, never one of each.

There are 35,960 possible ways to select 4 of the 32 tiles when the 32 tiles are considered distinguishable. However, there are 3,620 distinct sets of 4 tiles when the tiles of a pair are considered indistinguishable. There are 496 ways to select 2 of the 32 tiles when the 32 tiles are considered distinguishable. There are 136 distinct hands (pairs of tiles) when the tiles of a pair are considered indistinguishable.

Scoring[edit]

Each player groups their four tiles into two hands of two tiles each.^[6] The two hands are referred to as the "high" and "low" hands, based on their score.^[6]

The highest-ranked hands are formed from the sixteen named pairs.^[7] Otherwise, the next highest-ranked hand results from creating a Gong or Wong, which are specific combinations with the Day and Teen tiles. If the four tiles drawn for the two hands do not permit the formation of a named pair, Gong, or Wong, then the total number of pips on both tiles in a hand are added using modular arithmetic (modulo 10), equivalent to how a hand in baccarat is scored.

The name "pai gow" is loosely translated as "make nine" or "card nine".^[6] This reflects the fact that, with the exception of named pairs, Gong, or Wong, the maximum score for a hand of mixed tiles is nine.

Named pairs[edit]

Transcriptions

Pai Gow named pairs (Bo)^[4]^{: 204, 206}^[7]
Rank	Pips	Name
1 (highest)	4+2, 1+2	Supreme Gee Joon (至尊)

2	(6+6)×2	Heaven Teen (天)
3	(1+1)×2	Earth Day (地)
4	(4+4)×2	People Yun (人)
5	(3+1)×2	Goose Gor (鵝)
6	(5+5)×2	Plum Mooy (梅)
7	(3+3)×2	Long Chong (長)
8	(2+2)×2	Board Bon (板)
9	(5+6)×2	Ax Foo (斧)
10	(4+6)×2	Partition Ping (屏)
11	(1+6)×2	Seven Tit (七)
12	(1+5)×2	Six Look (六)

13	3+6, 4+5	Nines Chop Gow (紮九)
14	3+5, 2+6	Eights Chop Bot (紮八)
15	3+4, 2+5	Sevens Chop Chit (紮七)
16 (lowest)	3+2, 1+4	Fives Chop Ng (紮五)

The 32 tiles in a Chinese dominoes set can be arranged into 16 named pairs. Eleven of these pairs have identical tiles, and five of these pairs are made up of two tiles that have the same total number of pips, but in different groupings. The latter group includes the Gee Joon tiles, which can score the same, whether as three or six.

Any hand consisting of a pair outscores a non-pair, regardless of the pip counts. Named pairs are often thought of as being worth 12 points each, but there is a hierarchy within the named pairs.

The pairs are considered to tell the story of creation:

Gee Joon (至尊) is the highest ranked pair, and is the Supreme Creator of the universe
Teen (天) is the heavens, the first thing Gee Joon created.
Day (地) is the earth itself, placed under the heavens.
Yun (人) is man, whom Gee Joon made to live upon the earth.
Gor (鵝) is geese, made for man to eat.
Mooy (梅) is plum flowers, to give the earth beauty.

Each subsequent pair is another step in the story...robes (Bon) for man to wear, a hatchet (Foo) to chop wood, partitions (Ping) for a house, man's seventh (Tit) and eighth (Look) children.

Only the sixteen named pairs are valid. For example, if a hand contained a Yun (4-4) and a Chop Bot (3-5 or 2-6), these would not form a pair at all, despite both tiles having eight pips each. A Yun (4-4) exclusively pairs with the other Yun, and likewise only the two Chop Bot tiles can be paired together. Likewise, tiles with six pips (Look, 1-5, pairs with another Look, not the Gee Joon 2-4) and seven pips (Tit, 1-6, pairs with another Tit, not the two Chop Chit tiles 2-5 and 3-4, which pair with each other) are subject to the same pairing restrictions.^[7]

When the player and dealer both have a pair, the higher-ranked pair wins. Ranking pairs is determined not by the sum of the tiles' pips, but rather by aesthetics; the order must be memorized. The highest pairs are the Gee Joon tiles, the Teens, the Days, and the red eights. The lowest pairs are the mismatched nines, eights, sevens, and fives.

Wongs and Gongs[edit]

*Wong* and *Gong* combinations^[4]^: 207
	Wong (11)		Gong (10)
Teen (6-6)
	6-6, 5-4	6-6, 6-3	6-6, 4-4	6-6, 5-3	6-6, 6-2
	Teen, Gow		Teen, Yun	Teen, Bot
Day (1-1)
	1-1, 5-4	1-1, 6-3	1-1, 4-4	1-1, 5-3	1-1, 6-2
	Day, Gow		Day, Yun	Day, Bot

The double-one tiles and double-six tiles are known as the Day and Teen tiles, respectively. The combination of a Day or Teen with a nine (Gow, 5-4 or 6-3) creates a Wong, worth 11 points, while putting either of them with an eight (either Yun, 4-4; or Bot, 5-3 or 6-2) results in a Gong, worth 10. Gongs and Wongs formed with a Teen tile are ranked higher than those formed with a Day tile.^[8]

However, if a Day or Teen is grouped in a single hand with any other tile, the standard scoring rules apply. The combination of a Day or Teen with a seven (Tit, 1-6; or Chit, 2-5 or 3-4) is sometimes referred to as a high nine, as the score is the maximum (nine) when added together, and the group contains a high-rank tile for potential tiebreaking purposes.^[8]

Modular arithmetic[edit]

Basic scoring examples (modulo 10)
	Tiles		Total value	Score
Ex. 1			4+5=9	9
Ex. 1	1+3=4	2+3=5	4+5=9	9
Ex. 2			5+11=16	6
Ex. 2	2+3=5	5+6=11	5+11=16	6
Ex. 3			10+10=20	0
Ex. 3	5+5=10	4+6=10	10+10=20	0
Ex. 4			3+11=14 or 6+11=17	7
Ex. 4	1+2=3 (or 6)	5+6=11	3+11=14 or 6+11=17	7

When a hand is formed from two tiles that are not a named pair, Wong, or Gong, the total pips on both tiles are counted and any tens digit is dropped; the resulting ones digit (the sum of all pips modulo 10) gives the final score.^[4]^: 205

There is one exception. The 1-2 and the 2-4 tiles which form the Gee Joon pair together, can act as limited wild cards singly. When used as part of a hand of mixed tiles, these tiles may be scored as either 3 or 6, whichever results in a higher hand value.^[8] For example, a hand of 1-2 (scored as +6 instead of the face value of +3) and 5-6 (+11) scores as seven rather than four.

If the player has both the 1-2 and 2-4 tiles, those collectively form the highest-ranked named pair and should be used together to form an unbeatable rear hand.

Ties[edit]

When the player and dealer display hands with the same score, the one with the highest-valued tile (based on the named pair rankings described above) is the winner.^[4]^: 204 For example, a player's hand of 3-4 and 2-2 (Chit and Bon) and a dealer's hand of 5-6 and 5-5 (Foo and Mooy) would each score one point. However, since the dealer's 5-5 (Mooy) outranks the other three tiles, he would win the hand.

If both have a bonus combination (Wong or Gong) or the scores are tied, and if the player and dealer each have an identical highest-ranking tile, then the dealer wins.^[4]^: 205 For example, if the player held 2-2 and 1–6 (Bon and Tit), and the dealer held 2-2 and 3–4 (Bon and Chit), the dealer would win since the scores (1 each) and the highest-ranked tiles (2-2 Bon) are the same. The lower-ranked tile in each hand is never used to break a tie.

There are two exceptions to the method described above. First, although the Gee Joon tiles form the highest-ranking pair when used together, when used as single tiles in a mixed hand, for tiebreaking purposes, they fall into the mixed-number ranks according to the number of pips. That is, the 2-4 ranks sequentially below the Chop Chit tiles (3-4 and 2-5), and the 1-2 ranks sequentially last overall, below the Chop Ng tiles (3-2 and 1-4).^[4]^: 205 Second, any zero-zero tie is won by the dealer, regardless of the tiles in the two hands.^[4]^: 205

Strategy[edit]

Hand formation examples
Ex. 1	Hand 1A		Hand 1B		0,0
Deal					Score
Deal	A (2+2=4)	B (1+5=6)	C (5+6=11)	D (4+5=9)	Score


	A+B (4+6=10)		C+D (11+9=20)

Ex. 2	Hand 2A		Hand 2B		5,5

	A+C (4+11=15)		B+D (6+9=15)

Ex. 3	Hand 3A		Hand 3B		3,7

	A+D (4+9=13)		B+C (6+11=17)

The key element of pai gow strategy is to present the optimal front and rear hands based on the tiles dealt to the player. For any four random tiles, there are three ways to arrange them into two hands, assuming that a named pair cannot be formed. However, if there is at least one pair among the tiles, there are only two distinct ways to form two hands.

The player must decide which combination is most likely to give a set of front/rear hands that can beat the dealer, or at least break a tie in the player's favor. In some cases, a player with weaker tiles may deliberately attempt to attain a push so as to avoid losing the bet outright. Many players rely on superstition or tradition to choose tile pairings.

In popular culture[edit]

The film Premium Rush (2012) features Pai Gow play as an integral plot element.^[9]

References[edit]

^ Weirather, Larry (2015). Fred Barton and the Warlords' Horses of China: How an American Cowboy Brought the Old West to the Far East. McFarland. p. 41. ISBN 978-1-4766-2079-4.
^ "Pai Gow, China's Lost Legacy". China Underground. October 24, 2017.
^ "History of Pai Gow". Casino Daily News.
^ Jump up to: ^a ^b ^c ^d ^e ^f ^g ^h Helprin, Syd (1986). "15: The Oriental Games". The Gambling Times Guide to European & Asian Games. Secaucus, New Jersey: Lyle Stuart. pp. 194–209. ISBN 0-89746-062-6. Retrieved 25 September 2023.
^ Chan, Jackie; Yang, Jeff (1998). "High Risk". I Am Jackie Chan: My life in action. New York, New York: Ballantine Books. pp. 154–156. ISBN 978-0-345-42913-1.
^ Jump up to: ^a ^b ^c Yao, Pai (30 June 2012). "Pai gow part A: the tiles and basic play". World Gaming Magazine. Retrieved 22 September 2023.
^ Jump up to: ^a ^b ^c Yao, Pai (31 August 2012). "Pai gow part B: pairs and standard hands". World Gaming Magazine. Retrieved 22 September 2023.
^ Jump up to: ^a ^b ^c Yao, Pai (31 October 2012). "Pai gow part C: wongs, gongs, high 9s and gee joon". World Gaming Magazine. Retrieved 22 September 2023.
^ "'Premium Rush': The Greatest Pai Gow Movie of All Time". Hollywood.com. January 8, 2022. Retrieved 25 September 2023.

External links[edit]

Wikimedia Commons has media related to Pai Gow.

Pai gow at BoardGameGeek
Scoring chart
Pai gow lore at Wizard of Odds website (Michael Shackleford)
Musante, Michael J. (1983). Pai Gow: Chinese dominoes. Casino Press. ISBN 0-89650-815-3.

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The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12?

History[edit]

Rules[edit]

Starting[edit]

Scoring[edit]

Named pairs[edit]

Wongs and Gongs[edit]

Modular arithmetic[edit]

Ties[edit]

Strategy[edit]

In popular culture[edit]

See also[edit]

References[edit]

External links[edit]

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