The first playing cards appeared in China, when paper started to be used in sheets
and books, rather than rolls.
In 868, the
Collection of Miscellanea at Duyang
(by E Su)
says that the favorite daughter of
Emperor Yizong
of the Tang dynasty,
Princess Tongchang,
would play the game of leaves (yeh-tzâ)
with members of the family of her powerful husband
(Wei Baoheng).
In 1120, an official memo proposed a standardization of
the popular card game of "Heavens and Nines" (T'ien-kiu)
using a type of playing cards
known as dotted cards (teen tsze pae).
The
fallacious legend
that those cards were invented at that date to amuse
the numerous concubines of Emperor
Huizong
(of the Song dynasty)
was properly debunked by W.H. Wilkinson (1895) and others.
The legend that European explorers of the 13th and 14th century (starting with
Marco Polo)
brought playing cards from the Orient to Venice is dubious at best.
So are the reports that
Bertrand du Guesclin (1320-1380)
discovered them in Italy
"in 1350",
during the Hundred Years' War.
In 1371, the Catalan word naip was used to
describe a playing card (the Spanish spelling is naipe).
According to Michael Dummett
(1925-2011) that's the earliest extant European reference to playing-cards.
Cards became really popular in Europe around 1377,
at which time they attracted the attention of religious and civil authorities
(until then, dice were the sole gaming enemy).
In 1377, a gaming statute was passed in Florence to regulate the
"recently introduced" game of naibbe.
The most commonly encountered commercial sizes of playing cards are
tabulated below, together with rarely-used standard "B" sizes
(ISO 216).
Also included, for good measure, are the series of Archimedes
and Fibonacci sizes, which are only of theoretical interest at this time
(with the possible exception of the 62.5 by 100 mm format,
which is very close to the 62 by 100 size of some novelty decks by Cartamundi).
Cards of more than 105 square centimeters are considered
oversized.
Conversely, the surface area of miniature cards is less
than 35 cm2.
Oversized , Regular
and Miniature Card Sizes
(rounded corners of radius R)
(2013-06-16) How Playing Cards are Made
Two layers of paper (with dark glue) or 100% plastic.
For maximum opacity playing cards are made from two layers of paper,
bonded with black glue. After printing, a plastic finish is
applied which determines the feel of the cards
(and establishes the reputation of a brand).
The latest trend is to eliminate paper entirely and produce "100% plastic" cards
made from either
PVC or
cellulose acetate
(the later is heralded as feeling more like paper).
The better brands apply the same plastic finish
to their plastic cards as they do to their paper ones.
Before industrialization, French cards were obtained by using traditional
glue (flour and amidon cooked together) to bind three layers of paper:
Papier cartier : For the backsides, free from identifying defects.
Main-brune : Low-grade paper, providing thickness and opacity.
Papier au pot : Supplied by the taxing authorities, for card fronts.
(2013-05-21) The Four Suits
(French: Enseignes or Couleurs )
The modern designations came from ancient ones, tied to social classes.
The modern suit symbols (spades, hearts, diamonds, clubs)
in two colors (black and red) were introduced in France around 1480.
They are still known as French suits.
They were competing with the Latin suits
(swords, cups, coins and staves) and the German suits
(leaves, hearts, bells and acorns) of which the Swiss suits
were a major variant (shields, roses, bells and acorns).
Despite the prevailing political animosity between France and England at the time,
the French suits eventually spread all over Europe, because they
greatly simplified the printing of playing cards
(the Latin and German suits were printed in full colors).
In recent years, the traditional German suits were mostly favored by
East-Germans. Shortly after reunification (in the 1990's)
a compromise was adopted for the official decks of Skat
tournaments, which now have the French-design suits of West-Germany with
the traditional East-German colors
(the spades are green and the diamonds yellow,
as reminders of the leaves and bells, respectively).
The game of Tarot features trump cards that
do not belong to any suit.
Trump cards make trick-taking card games more interesting
but dedicated ones increase the size of a deck
(cost considerations were not insignificant in the old days).
Some of the best card games
(bridge, belote, skat) are based on the idea that one of the four
regular suits could be used as trumps after a preliminary negotiation
stage (bidding, enchères) once
the players have received their cards.
This idea was introduced in the game
Ombre
(2013-05-26) Court Card & Face Cards (or figures )
If there are only 3 face cards per suit, court cards may include aces too.
Originally, the highest-ranking cards in a suit were the four
face cards: king, queen, knight and jack. Aces were the
lowest-ranking members of a suit. This remains true in tarot decks.
At least one modern tarot deck (The Alchemical Tarot
by Robert M. Place) has two female figures in the court:
King, Queen, Knight and Lady.
For tarot packs, the locutions court cards and
face cards are synonymous. In decks containing only
three face cards per suit, however, the aces are normally considered to
rank above the king and they belong to the court.
Thus, there are always four court cards.
In most decks with three face-cards, the missing face card is usually one of
the two knaves (the remaining one is dubbed
jack in modern English).
A crown identifies the king (K for König ).
Next is the cavalier, riding a horse, called Ober (O).
Last is the so-called
Unter (U or B for Bube ).
Modern German decks follow the international tradition by retaining the king (K)
the queen (D for Dame) and only one knave,
called Bube or Bauer (B)
sometimes also known as Junge or Wenzel.
The first card ladies were French :
The earliest extant record of
female card figures
dates back to 1392,
when the painter Jacquemin Gringnonneur delivered to Charles VI
three decks he had been commissioned for.
In those decks, the cavaliers were replaced by queens.
This was before the invention of the tarot deck, which has
both cavaliers and queens.
The royal treasurer, Jacques Poupart, paid
Gringnonneur 56 sols parisi for his work.
This was apparently all it took to set the French pattern of 3 face cards which
is dominant today.
(2013-05-27) 52-card Deck
The Mameluke deck (c. 1370)
begat the modern English
packs for Bridge and Poker.
The oldest full 52-card deck was identified
in November
1983, by an Amsterdam dealer who got it in 1978 for less than $3000.
Those large oval cards were manufactured in Flanders between 1470 and 1479
(probably in Lille, modern-day France) from paper made before 1450.
The deck has three figures (Kings, Queens, Knaves)
all hand-painted in three-quarter pofiles. The four suits
have a hunting theme:
(2013-05-22) The Major Arcana (c. 1440)
What occultists call the 21 numbered trumps and the fool.
The trumps in the tarot deck are a European
invention inspired by the works of the Italian poet
Petrarch (1304-1374)
who started writing I Trionfi after 1352
(his manuscript was printed posthumouly in Rome, in 1471).
Petrarch's triumphs (that's where the word trumps come from)
include the triumph of Chastity over Love and Love over Mankind.
Eternity triumphs over Time, Time over Fame, Fame over Death, Death over Life.
The charade was pseudo-knowledge that was probably commonplace when the
tarot was designed. It served as little more than a graphic mnemonic.
With the passage of time, the legend faded away but the cards remained,
acquiring an aura of mystery which was never meant...
The other 56 cards (Minor Arcana)
are divided into 4 suits
(swords, cups, coins and staves)
each with 4 court cards:
king, queen, knight and knave.
(2013-05-30) French Names of Court Cards (c. 1460)
Joan of Arc lives on, thinly disguised, in French decks of cards. Un étendard blanc semé de lys, avec un monde entre deux anges.
In Europe then, news spread apace and unlettered folk got to know, in some strange way, the doings of camps and courts.
Virginia Frohlick
about Joan of Arc
It's been exactly 582 years since the infamous day
(May 30, 1431)
when a teenager was burned at the stake for witchcraft, because
she heard voices and answered their call. She had led armies into battle on horseback and
secured the throne of the French king, at the beginning of the end
of the
Hundred Years' War
(1337-1453). She was Joan of Arc, she is Queen of Spades.
Playing cards appeared in France
around 1377,
a few months before Charles VI, le fol,
would begin his 42-year reign at age 11.
The court cards would receive their formal names
toward the end of the reign of his disinherited son,
Charles VII
(1403-1461) who was mocked as le petit roi de Bourges
until his grand coronation in Reims (1429)
which was brought about by the fabulous efforts of Joan of Arc,
La Hire, Xaintrailles et al.
The dominant card game of the era was Piquet.
A dubious legend attributes that game to the knight
who gave his name to the jack of hearts, LaHire, Etienne de Vignolles (1390-1443)
nicknamed La Hire-Dieu (the Wrath of God) by friends and foes alike.
In Constant Leber's compilation
(1838) the Père Daniel
construes the rules and strategies of piquet to be
reminiscent of the reversals of fortunes during the
Hundred Years' War. Daniel also claims that Hector, the jack of diamonds,
stood for Hector de Galard,
although Jean-Baptiste Bullet
(1699-1775) had already positively identified the jack of diamond as the
Homeric hero Hector of Troy, one of the nine worthies
(a list of 9 all-time heroes, made in 1312, which remained quite popular for about three centuries).
Jeanne d'Arc (Joan of Arc) lives on as the queen of spades,
dame de pique or, appropriately,
Lady of Swords. The printed name is Pallas
(Athena, the celibate goddess of war) a phonetic anagram for pucelle
(the virgin) Jeanne's popular nickname
(used in her ennoblement decree of 1429).
The names still printed on modern French 32-card decks
form a riddle which would have been easy to decode in 15th-century France.
The playing cards were the tabloids of that era and the names of the
celebrities were thinly disguised with anagrams
(Argine = Regina = the reigning queen) or phonetic inversions
(Pallas = Pucelle = Joan of Arc, dame d'épée).
Naming three of the kings after legendary rulers certainly
flattered the fourth, the reigning king Charles VII, king of cups.
The popular interpretation of Charles as
Charlemagne (742-814) probably came only much later
(that made the 4 card kings belong to the Nine Worthies).
The four ladies are directly connected to Charles VII.
They are respectively his mother, his wife, his official mistress and finally,
last but not least, the aforementioned lady of swords (Pallas = Pucelle = Virgin)
who helped secure his throne: Joan of Arc,
burned at the stake at the age of 19 (1431) after a year of incarceration and a rigged
trial.
She was innocented posthumously in 1450 and fully rehabilitated by the Church
in 1455.
Jeanne d'Arc
(1412-1431) was ennobled by Charles VII in
December 1429.
In an unprecedented move, the king granted nobility and tax-exempt status to
Jeanne's entire family (her father Jacques, her mother Isabeau, her
three brothers Jacquemin, Jean & Pierre) and all their descendants through
the male and female lines (this was partly revoked in
1614).
Her family would then use the noble name "du Lys" (now "Dulys").
During WWII, Charles de Gaulle (1890-1970)
used a famous symbol linked to Jeanne d'Arc, the Cross of Lorraine,
as the symbol of Free France against nazi occupation.
(Jeanne herself only sported a white
bocassin banner, semy of lys Or,
an orb between two angels, next to the words Jhesus Maria.)
I had a banner of white bocassin, fringed with silk.
The field was sprinkled with lilies,
with an orb painted between two angels.
On the side was written, I believe, "JHESUS MARIA".
Jeanne d'Arc, at her trial (1431)
Hector de Galard belonged to the court of Charles VII and was the founding captain, in 1474,
of the Grande Garde, an elite military unit of 100 knights
formed by Louis XI (son, foe and successor of Charles VII)
for the protection of the king (nicknamed
gentilshommes à bec-corbin, because of their weapon of choice).
Galard's identification with the jack of diamonds isn't nearly as firmly established as
the identification with the jack of hearts of La Hire
(Etienne de Vignolles) the most prominent military leader in the campaigns of
Joan of Arc,
who died pauvre et glorieux at Montauban on January 12, 1443.
Actually, Hector was almost certainly intended to be
Hector of Troy, one of the so-called
Nine Worthies
(les neufs preux) who had been singled out
by Jacques de Longuyon
(1312) as personifying the medieval ideals of chivalry.
That popular list goes a long way toward
explaining how the male names of the court cards were chosen, about 150 years later.
The Nine Worthies (les neufs preux) after Jacques de Longuyon (1312)
Hogier, the jack of spades, was thus meant to be
Hoshe'a, son of Nun (whom Moses called Joshua, cf.
Numbers 13:16).
The identification to
Ogier the Dane
isn't supported by anything, except similar spelling !
The Last Court Card : Valet de Trèfle
The jack of clubs is now dubbed Lancelot.
It was originally dedicated to
Judas Maccabeus,
another one of the nine worthies,
but the name of Judas
was supposedly omitted because of the embarrassing homonymy with
Judas Iscariot,
the betrayor of Christ (in French, "Judas" is a
common synonym of "traitor").
French card masters would sign their names on the jack of clubs
or put their marks on the shield carried by that jack.
In the 1600s, one of them didn't print anything distinctive
on his jack of clubs.
Strangely, he became known by the standard
inscription he put on the jack of diamonds :
Ector de Trois.
When card makers started to sign one of the aces instead,
it was natural to give a new name to the jack of clubs.
An obvious choice was the most famous of all knights:
The fictitious Lancelot du Lac,
chief knight of King Arthur's Round Table,
lover of Queen Guinevere
(Guenièvre)
and seeker of the Holy Grail.
This very popular character was first introduced by
Chrétien de Troyes
in the romance of
"Lancelot,
le chevalier à la charrette" (c. 1172).
(2013-05-29) Jokers (1857)
The 25th card in Euchre / Jucker was called best bower.
Although traditional card games are played without jokers,
virtually no modern 52-card deck
is sold without at least one extra joker card
(usually two, sometimes four).
Jokers are primarily used for casual poker games, which are always
played with a 52-card deck.
In the old game of Jucker from
Alsace,
the two most powerful cards are two jacks of the same color
(called Juker, regionally, or Bauer
in German). When that game was exported to the United States,
its name was distorted to Euchre and
Bauer became Bower (both spellings
approximate the German pronunciations). The game was originally
played with a deck of 24 cards (it still is).
An American innovation was to introduce one extra card, called
best bower that would take either bower...
This became known as the Jucker card.
The joker was born.
The first joker was made for the London Club Park deck,
published in 1857 by Samuel Hart (New-York).
The joker can also be construed as a descendant of the ancient
fool from the tarot deck
(which plays a key rôle in the tarot game, as one of only three oudlers ).
In most games the joker plays the same rôle as that original fool
(a wild card that can take the place of any other card, as the need arises).
The first French joker :
Jokers were only officially allowed into French decks in 1902,
at which time the illustrator
Henri Bellery-Desfontaines
(1867-1909) designed
the first French joker
for a deck published by Fossorier Amar et Cie,
supposedly as a portrait of the "royal dwarf"
Haincelin Coq
(also spelled Hainselin).
Every detail in the picture (below) seems wrong :
Coq's costume was mostly green with some red elements.
The bonnet à grelots
( fool's cap with bells)
wasn't worn by court jesters before the end of the 15-th century.
Finally, the cunning look seems to betray an intelligence which, alas,
Haincelin Coq didn't possess at all...
Haincelin Coq was
fool
to Charles VI, the mad, for most
of his disastrous 42-year reign (1380-1422).
Coq once tore his clothes into shreds while leaping and dancing before the king.
His trademark garment was a green bastard houppelande
which became known as haincelin
after him (green was the color of court jesters).
He wore out an inordinate number of shoes
(the record shows that he received 47 pairs in 1404).
Haincelin Coq was cared for by a varlet
(Jehan Faucon in 1387,
Jacquet Coiffar in 1404,
Perron Ducreux in 1407).
Hainselin Coq became fool to Charles VI in
1383
(Grand Jehan, who had been the future king's personal fool since 1374, had died in 1382).
Coq was probably a young child at the time, dim and difformed.
He survived Charles VI.
Joker (Wikipedia)
|
Joker by Victor Mauger
(New-York, 1875)
|
Origin of Jokers by Tomasz Czapla
Jeanne et le feu by Matei Visniec
(theatrical play, 2007). Jesters of all courts "create" Joan of Arc.
An international network of joker collectors exists, with cross-linked
amateur websites displaying private collections and/or offering duplicates
for trade.
Dozens of such sites are online, as of June 2013, including:
Italy :Guiness World Record (2009)
Donato "Tony" de Santis ammassed 8520 different jokers, as of 2009.
Tony built upon a collection of about 2000 inherited in 1999 from his mentor, Fernando Riccardi.
(2013-06-05) 48-card aluette deck, lacking the 10 (c. 1530)
The mimicks of Aluette are tied to Latin suits and traditional designs.
Nowadays, packs of Mexican origin are commonly sold as Spanish decks of "50 cards"
(48 cards plus two modern jokers, called comodines)
in the proper Latin suits (called espadas, copas, oros, bastos)
but without the popular symbolism associated to the game of Aluette,
which seems to have been limited to the west coast of France
(mostly, from cardmakers located in the regional capital of
Nantes).
(2013-06-18) Skat, the quintessential 3-player card game.
Created in
Altenburg around 1815,
it's now part of German culture.
To this day, part of the economy and cultural identity of Altenburg revolves
around Skat. The city is known far and wide as Skat's town
( Skatstadt ).
Three players play the game, using a 32-card deck.
The dealer deals "3-skat-4-3" clockwise after shuffling the deck.
That's to say, he first gives 3 cards to the player to his left, 3 cards to the third
player, 3 cards to himself and discards 2 cards (the Skat )
in the middle of the table. He then serves a round of 4 cards and a final round of 3
cards, so everyone has 10 cards at the beginning of the bidding.
The 2 cards of the Skat will belong to the winner of the auction, the declarer, who will play
10 tricks against the other two, aiming either to win most trick points
or none at all (the latter applies when a Null game is declared).
The player who wins the auction by biding the highest number chooses
the type of play and the trump color, if any
(the 4 jacks are the highest trumps, except in a null game).
He then plays against the other two for a score computed according to his
12 cards (10 original cards plus the Skat ).
(2013-05-25) Happy Families (John Jaques II, 1851)
Deck of 44 cards. Eleven families (father, mother, son, daughter).
The game was launched by Jaques & Sons at the
Great Exhibition of 1851.
The 4 cards in every family are labeled after the pattern:
Mr. Pots, the Painter.
Mrs. Pots, the Painter's Wife.
Master Pots, the Painter's Son.
Miss Pots, the Painter's Daughter.
The 11 original families were:
Block, the Barber
Bones, the Butcher
Bun, the Baker
Bung, the Brewer
Chip, the Carpenter
Dip, the Dyer
Dose, the Doctor
Grits, the Grocer
Mug, the Milkman
Pots, the Painter
Soot, the Sweep
Besides the funny names tied to the professions,
the initial appeal of the game was due in no small part to the
44 caricatures by John Tenniel (1820-1914)
the chief cartoonist for
Punch.
The deck underwent many minor revisions,
starting with the replacement of "Mug, the Milkman" by
"Tape, the Taylor" (before 1880).
The game can also be played with a regular deck.
There are two main variants; the first one is normally played
with a full deck (52 cards, 13 families) and the second
one with lesser decks, including the 44 cards of traditional
specialized Happy Families decks:
Go Fish! Hunting for a whole family:
"Do you have any kings?"
Quartets Hunting for a card:
"Do you have the king of clubs?"
Merchandising vs. Education :
Modern merchandisers often shortchange unsuspecting parents by peddling smaller
decks of only eight families (32 cards)
based on the latest craze.
On outter packages, there's no hint of the swindle, which is made
even easier by the lack of proper labelling on traditional 44-card decks.
Another unfortunate trend is the substitution of codes (A1, A2, A3, A3, B1, B2, etc.)
for the words that formerly identified each card.
The lack of formal sentences
during the game is a definite loss for younger players...
Conversely, although they tend to be pricey,
I recommend the luxury educational decks of 48 cards (12 families) that
forgo codes and patterns entirely: Each oversized card
(70 mm by 120 mm) illustrates a topic and quotes
the titles of the three cards it forms a "family" with.
Modern:
Papy,
Mamie, Papa,
Maman,
Grand frère,
Petite sœur.
In 1875, a few months before he put together the first jeu des sept familles
the famous caricaturist André Gill painted
a famous sign, of a rabbit (lapin) jumping out
of a saucepan, for a cabaret in Montmartre which became known as
Le lapin à Gill (Gill's Rabbit) then
Le lapin agile
(The Nimble Rabbit). It was notoriously patronized by
struggling artists and writers before they achieved great fame,
including Courteline, Max Jacob, Marcel Proust,
Apollinaire, Modigliani, Renoir, Utrillo and Picasso.
The cabaret was bought by Aristide Bruant in 1903 and remains a quaint Parisian attraction.
(2013-05-24) French Mille Bornes
boardless game (Dujardin, 1954)
106 cards of 4 types (18 attaques, 38 parades, 4 bottes, 46 étapes).
Over 10 million of these automobile-themed games have been sold.
Attaques Attack
Parades Parade
Bottes Safety
3
Panne d'essence
6
Essence
1
Citerne
3
Crevaison
6
Roue de secours
1
Increvable
3
Accident
6
Reparation
1
As du volant
4
Limite de vitesse
6
Fin de limite
1
Véhicule prioritaire
5
Feu rouge
14
Feu vert
Etapes Progress
4
200
12
100
10
75
10
50
10
25
Under the pen name of Edmond Dujardin, the deaf Frenchman
Arthur Dujardin
(19??-1964)
was a publisher of driving-school material.
In 1949, he received a Concours Lépinesilver medal for designing a board game around an automobile theme,
called L'Autoroute.
Building on that first success, Dujardin went on to design the megaseller Mille Bornes
in 1954, inspired by the
Touring game of
William Janson Roche (1906)
with the key addition of safety cards and coup-fourré play.
The standard graphics of the 1000 bornes cards date back to 1960
and are due to the graphist
Joseph Le Callennec (1905-1988).
They appear on a bilingual edition introduced internationally in 1962.
About 200 000 Mille Bornes decks are sold every year.
A player is only allowed to play the highest distance denomination
(200 km) once per game. On the corresponding card (above right)
Le Callennec drew a swallow which is discreetly reminiscent
of the famous logo he had revamped for
La Pie qui Chante
candies ("Singing Magpie" at left).
Arthur "Edmond" Dujardin applied for a British patent on November 30, 1961
(GB963821-A, dated 1964-07-15).
Curiously, that patent doesn't describe the classical 106-cards deck but an augmented pack
of 112 cards, featuring
two "ace of the wheel" cards (instead of one) and a new category of five
"precedence cards" entailing bonus points and the privilege to draw twice, except when
another player reveals a higher-ranking precedence card...
Most cards shown in the patent are just black-and-white versions of
Le Callennec's artwork (1960).
However, the five additional cards are in a different style
(Fig. 16-20 on the British patent) :
300 points
200 points
150 points
100 points
50 points
That version was apparently never published. In the end, the extra feature may have been
too intricate (even if the number of bonus points had been printed on the new cards).
Note that 112 cards is a magic number using industry-standard printing
equipment, as it corresponds to two
standard plates of 56 cards (8 by 7).
Brian Kieffer
(2011-06-17)
Marsha Falco's SET® cards (1974)
What's the probability of having no valid "set" among 9 random cards?
The modern SET® playing cards form an 81-card
deck with a regular ternary structure, used to play a totally new species of games.
It's also a good pretext to practice some challenging combinatorial calculus.
The SET® cards were invented in
1974
by the Cambridge population geneticistMarsha J. Falco
who drew her inspiration from the visual filing system she had invented for herself
(in the course of her investigation of the hereditary causes of epilepsy in
German sheperds).
She copyrighted the game in 1988.
Since 1991, SET decks have been published
by Set Entreprises, a family business
now owned and operated by Colette Falco, daughter of Marsha and Robert Falco.
Each card is uniquely identified by the four following 3-valued attributes.
The 81-card deck covers all possibilities once and only once
(81 = 34 ).
Number : one, two or three (identical) symbols.
Shape : squiggles, diamonds or ovals.
Color : red, green or purple.
Shading : solid, striped or open (a.k.a. "empty").
By definition, a valid set consists of 3 cards which,
for every attribute, have either identical values or pairwise distinct ones.
To put it in a nutshell, three cards do not form a
"set" when "two are of one kind and the third isn't".
For any pair of cards, there's one and only one third card
that will form a valid set
(since only one value of each attribute isn't ruled out).
[ DEMO ]
Thus, a valid set is determined by two of its cards
in three different ways. So, there are
C(81,2)/3 = 1080 distinct valid sets.
Unless n = 3, we can't use this approach to analyze the
generalized game that uses a deck of nk cards bearing
k attributes having n possible values.
Instead, we count ordered SETs directly
(there are n! of those per SET):
Every ordered SET corresponds uniquely to k
sequences chosen among (n! + n) possibilities
(namely, n! permutations and n constant sequences,
assuming n > 1)
except that a choice of a constant sequence for all
attributes at once is disallowed (that woukd yield n identical cards).
All told, the number of n-SETs [ consisting
of n cards bearing separate attributes that are either all alike
or all different ] is thus equal to:
[ (n! + n) k - nk ] / n!
The probability that a hand of n cards forms a SET is
equal to that integer divided by the total number of possible hands,
namely C(nk, n).
Probability Pn,k (m)
of at least one SET among m cards :
Pn,k (m) is zero 0 when m < n.
Pn,k (n) =
[ (n! + n) k - nk ] /
[ n! C(nk, n) ]
is tabulated next.
Pn,k (m) is challenging to compute
when m > n. See below.
Probability that n cards with k (n-valued)
attributes form a SET
Pn,k (n)
n = 2
n = 3
n = 4
n = 5
k = 1
1
1
1
1
k = 2
1
1 / 7
8 / 455
13 / 5313
k = 3
1
1 / 25
19 / 13237
21 / 302621
k = 4
1
1 / 79
80 / 546227
313 / 120320613
k = 5
1
1 / 241
2801 / 177310271
521 / 5071628121
k = 6
1
1 / 727
6536 / 3812158805
7813 / 1906250203113
k
1
1 / (3k-2)
(7k-1) /
(4k-1)(4k-2)(4k-3)
(5k+1) /
(5k-2)(5k-3)(5k-4)
The classical SET® game (n=3, k=4)
corresponds to the highlighted entry in the above table.
The simple formula at the bottom of the column n=3
confirms our previous remark that, with 3-valued attributes,
any pair of cards is turned into a SET by one and only one of the remaining cards.
To design actual cards with k = 5 or 6 attributes with
n = 3,
we could add one or two independent features at the borders
of standard SET cards:
Edge : dark, light or white.
Corner : angle, square or round.
However, that would probably make the game too difficult to play because of the
large size of the decks (243 or 729 cards) and the great number
of cards that will typically be drawn before a SET is found...
On the other hand, k=2 (9 cards)
is too simple to be interesting.
Only k=4 and k=3 are actually played
(the latter mostly by beginners who use one third of a standard SET deck, like
the 27 red cards or the 27 solid ones).
With n = 3, two distinct SETs can have at most one card in common.
So, if a hand of m = 4 cards contains a SET, it contains only
one and is uniquely determined by the juxtaposition of a SET and one other card
(if k > 1):
4 cards yield C(4,3) = 4 mutually exclusive opportunities to form a SET.
With m = 5 cards, the similar enumeration of choosing a SET and a pair would count twice
the hands consisting of two SETs sharing one card.
Every such combination is obtained in 4 distinct ways by choosing one SET,
one of the 3 cards in that set, one random card and the final card that
makes a SET with the chosen card and the random one.
By inclusion-exclusion:
With m = 6 cards, let's first count the number of configurations that
consist of two disjoint SETs. This is equal to the number of pairs of
SETs minus the previously enumerated number of such pairs that share one card,
namely:
As usual, the simplicity of that result suggests
a better enumeration:
We may obtain two disjoint SETs in 2x3! = 12
different ways as follows:
Choose a SET and any other fourth card. As fifth card, only 3 cards from the rest of the pack are
disallowed (as they would form a SET with the fourth card
and one of the other three). The sixth card must be the one that forms a SET with the
fourth and fifth one. Done.
Now, a hand of 6 cards may also consist of 3 SETs that share a card pairwise.
Each such hand can be obtained in 3 different ways by choosing a SET, two cards in that SET
and one random card outside the SET (the two remaining cards are
those which form a SET with that card and either of the two previous choices).
Alternatively, as every pair of the 3 shared cards determines one unshared card,
there are as many of those special 6-card hands as there are combinations of 3 cards
that do not form a SET, namely:
It's impossible to have 4 SETs with 6 cards (every card being in 2 SETs)
because, in the only possible configuration at right
(where SETs are represented by aligned nodes) two nodes of the same color
must have the same value for every attribute
(fun to prove).
Thus, only identical cards could be placed
at two such nodes, which is ruled out with a single deck.
So, by inclusion-exclusion, we obtain the following expression
for P3,k (6):
To obtain the new deck number of a card,
add the weights of its various attributes to the number of objects on it.
This gives you a number from 1 to 81
(if you'd rather number such things from 0 to 80, either
subtract 1 from that total or assign a score from 0 to 2 to the number of objects).
Card 26 (2+6+18+0)
Card 56 (2+0+0+54)
Card 41 (2+3+9+27)
A new deck comes in two separate packs: A small pack with cards 1 to 27 in that order
(namely, single solid purple squiggle face up on top and triple solid red oval at the
bottom).
The rest of them (cards 28 to 81) are in the second pack
(single striped purple squiggle face up on top, and triple open red oval at the bottom).
For the record, on top of the big pack is a card with the coordinates of Set Enterprises, Inc.
(with instructions to obtain a replacement card on the reverse).
The publicity card on top of the small deck advertises 6 games starting with
traditional SET.
The packaging might gives some weak clue that the manufacturer originally intended
the large pack to be "first" (they put it in the left compartment of
the box when the publicity cards are right-side up).
However, far more importantly, their advice is to play
only with solid cards when a reduced deck of 27 is desired
(for geginners and/or young children). That's why it's best to assign numbers 1-27
to the "solid" pack (this gives compatible numbers to the reduced pack and the
full deck).
(2015-03-01)
Zener Cards (ESP cards)
Five symbols: Round (1), cross (2), waves (3), square (4), star (5).
The five cards pictured above were designed in the early 1930s by the
psychologist Karl Zener (1903-1964) to help
in parapsychological investigations by J.B. Rhine of
extrasensory perception
(clairvoyance or telepathy).
Only the black-and-white version was ever widely used.
A standard Zener deck consists of 25 black-and-white Zener cards
(5 copies of each). The person to be tested is supposed to
guess the cards face-down (clairvoyance) or while someone else looks
at them (telepathy).
Any statistically-significant deviation from a success rate of 20%
would be meaningful
(counting either the overall hit rate or the rate for each type of card called, since
the rate for each type of card presented is very easily biased by personal preferences).
No such thing has ever been observed under
controlled conditions where all forms of cheating and/or
sensory leakage are ruled out.
(E.g., card counting, stacked deck, marked cards,
reflections off the glasses/eyes of the examiner, etc.)
If anything, the use of such cards has now become a joke.
Marked decks of Zener cards may well be more common than honest ones!
I had a banner of white bocassin, fringed with silk.
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