We close out Games and Decision Theory with The New Eleusis. The game Eleusis was invented by Robert Abbott, then an undergraduate at Colorado University, in 1956. Abbott had been studying the “Aha!” moment when people make discoveries, and Eleusis, named after the ancient Greek Eleusinian Mysteries, is a good simulation system for this. Abbott released the full rules for the game in his book Abbott’s New Card Games (1963). Martin D. Kruskal, a mathematical physicist, made improvements to it, releasing his rules in the monograph, Delphi: A Game of Inductive Reasoning (1962). Gardner himself introduced Eleusis in his column in Scientific American in 1959, and because it had evolved so much over the years, he decided to write the current chapter as an update.
For the newest (at the time of the writing for this chapter) version, you need 4 to 8 players, and two standard decks of playing cards (sometimes a round lasts long enough that you need a third deck). A full game consists of one or more rounds, and players take turns dealing each round. Because the players are attempting to divine the thinking processes of the dealer, that player is often given the titles God, Nature, Tao, Brahma, the (Delphi) Oracle, or simply Dealer. The dealer’s first job is to think up a “secret rule.” This rule defines what cards can be legally played during a player’s turn. The players have to determine this secret rule, and the faster one of them does, the higher their score. One key part of the game is that the dealer gets scored too, so it’s to their benefit to create a rule that is not too easy or too hard.
An example of a rule that’s too easy: “Play a card of a color different from the color of the last card played.” Gardner’s suggestion for a better rule is: “Play so the primes and non-primes alternate.” But, depending on the players, this rule could be too hard or too simple. A rule that’s too difficult would be: “Multiply the values of the last three cards played and divide by 4. If the remainder is 0, play a red card or a card with a value higher than 6. If the remainder is 2, play a black card or a picture card… etc.”
Martin offers 3 example rules for use with inexperienced players.
1) If the last legally played card was odd, play a black card. Otherwise, play a red one.
2) If the last legally played card was black, play a card of equal or higher value. If red, play a card of equal or lower value (face cards are numbered 11, 12, 13. Ace = 1).
3) The card played must be either of the same suit, or the same value as the last card played.
“The secret rule must deal only with the sequence of legally played cards.” It cannot include things like the sex or ages of the players, the time of day, or whether Dealer scratches his/her ear, etc. The secret rule must be written down for later confirmation, and it is ok for Dealer to give a hint, such as “Suits are irrelevant to the rule,” or “The rule depends on the two previously played cards.”
After recording the rule, Dealer shuffles the double deck and deals 14 cards to each player and none to themselves. They then place a “starter card” face up on the table to the extreme left of the playing surface. To determine the starting player, Dealer counts off the value of the starting card, going around the players from Dealer’s left. The indicated player begins play, which continues clockwise around in the circle. A play consists of placing one or more cards on the table. To play a single card, the player takes a card from their hand and reveals it to everyone. If the card is playable according to the rule, Dealer says “Right,” and it is then placed to the right of the starter card on the “main line.” (The main line is the list of correctly played cards extending horizontally to the right. See below.)
If the card doesn’t fit the rule, Dealer says “Wrong,” and it’s placed under the last card played. Vertical columns of cards are called “sidelines,” and consecutive incorrect plays extend the same sideline downward. Dealer gives that player 2 more cards, expanding their hand, as a penalty. If a player thinks they know the rule, they can play a “string” of 2 to 4 cards at once, overlapping the cards slightly to preserve their order and to show them to everyone. If the full string is correct, Dealer says, “Right,” and they are added to the main line as normal. If any one card is wrong, Dealer says “Wrong” to the entire string, and deals out twice as many penalty cards as there are cards in the string. The below illustration shows an example of all the rules explained so far. The secret rule is: “If the last legal card is odd, play black, otherwise play red.”
(All rights belong to their owners. Images used here for review purposes only. Example New Eleusis game in progress.)
Players get better scores by getting rid of as many of their cards as possible, and it helps to guess the secret rule to do that. If a player thinks they know the rule but they don’t have any further cards to play, they can declare “No play” and show all their cards to everyone. If Dealer says, “Right” and the player has 4 or fewer cards left then the round ends. If the player has 5 or more cards and is right, their cards are returned to the deck and they are re-dealt 4 fewer cards than they’d been holding and the round continues. If the Dealer announces “Wrong,” Dealer takes one of the correct cards and puts it on the main line, and deals that player 5 more cards as a penalty. Therefore, if a player hasn’t figured out the rule but thinks they have no legal play, it’s better if they just play a card at random instead of saying “No Play.”
Alternatively, if the player thinks they know the secret rule, they can declare themselves a Prophet under the following conditions:
1) They have just played (correctly or incorrectly) and the next player hasn’t played yet.
2) There’s no Prophet yet.
3) At least two other players besides themselves and the dealer are still in the round.
4) They have not been a Prophet before in this round.
When a player declares themselves to be a Prophet, they put a marker on their last card played (a black chess king or queen). They keep their cards but stop playing them until they are overthrown. The play continues clockwise, skipping the Prophet. When a player plays a card, the Prophet says “Right” or “Wrong”, and Dealer validates or invalidates the call with “Correct” or “Incorrect.” If correct, the played cards are placed on the main or side lines as before. If the Dealer says “Incorrect,” the Prophet is immediately overthrown and declared a False Prophet. Dealer removes the False Prophet’s marker from the table, and deals out 5 extra penalty cards. The False Prophet cannot become a Prophet again in that round, but any other player can. Abbott has said there’s an amusing analogy with science here: “The Prophet is the scientist who publishes. The False Prophet is the scientist who publishes too early.” Martin adds “It is the fun of becoming a Prophet and overthrowing a False Prophet that is the most exciting feature of New Eleusis.”
After a Prophet fails, Dealer takes over as Dealer again, completing the play that overthrew the Prophet, placing the card or string on the main or side lines. If the play was wrong, Dealer does not give the last player any penalty cards. This exemption rewards players for taking daring plays to overthrow the Prophet. In the words of Karl Popper, it “encourages scientists to think of ways of “falsifying” a colleague’s doubtful theory.”
Things get a bit complicated if there is a Prophet and a “believer” declares “no play.” If the Prophet says “Right,” or the Prophet says “Wrong” and Dealer says “Incorrect,” play continues as described above. However, if the Prophet says “Wrong” and Dealer says “Correct,” then the Prophet has to select what they think is a correct card from the player’s hand. If they do this right, then they deal out 5 penalty cards to the current player and play continues as normal. If they pick the wrong card from the player’s hand, they’re overthrown, and Dealer takes over again as before.
If thirty cards have been played and there’s still no Prophet, players get expelled from the round if they make a wrong play. The expelled player still receives penalty cards for the wrong play, for scoring purposes. If there is a Prophet, then expulsions are delayed for at least 20 cards after the Prophet’s marker. Pawns are used to show that expulsion is possible; if there is no Prophet then white pawns are set down every 10 cards on the layout. If there is a Prophet, black pawns are placed every tenth card following the Prophet’s marker. When a Prophet is overthrown, the black pawns are removed with the Prophet’s marker.
For scoring:
1) The greatest number of cards held by anyone is called the “high count.” Each player (including the Prophet if there is one) subtracts the number of cards still held in their hand from the high count, and the difference is their score. Players with no cards get a bonus 4 points.
2) The Prophet, if there is one, gets a bonus of the number of main line cards that follow their marker, plus twice the number of sideline cards after their marker. I.e. – one point for each correct card, and two points for each incorrect one.
3) The dealer’s score is the highest score of any player. However, if there is a Prophet, count the number of cards (right and wrong) preceding the Prophet’s marker and double this number. If the number is smaller than the highest player score, use the smaller number.
If you have time for another round, then you choose a new dealer, and hopefully everyone at the table will get their turns. However, some rounds can take several hours, so when you finish playing, add up all the scores at the end of the rounds played and give 10 points to anyone that hadn’t been a dealer as compensation.
(Full sample round, left half.)
(Full sample round, right half.)
In the above game, we have the main line and 9 side lines. White pawns are used to mark every tenth card. This was a 5-player game. Martin’s example is: “Smith was the dealer. The round ended when Jones got rid of her cards. Brown was the Prophet and ended with 9 cards. Robinson was expelled when he incorrectly played the 10 of Spades; he had 14 cards. Adams had 17 cards at the end of the game. The high count is 17. Therefore Adam’s score is 17 minus 17, or 0. Robinson’s score is 17 minus 14, or 3. Jones receives 17 minus 0, or 17, plus a 4-point bonus for having no cards, so her score is 21. Brown gets 17 minus 9, or 8, plus the Prophet’s bonus of 34 (12 main line and 11 sideline cards following his marker), making a total score of 42. This is the highest score for the round. Twice the number of cards preceding the Prophet’s marker is 50. Smith, the dealer, receives 42 because it is the smaller of the two numbers 42 and 50.”
In the addendum, Martin mentions two new games that came after Eleusis and Delphi – Sid Sackson’s Patterns, and Robert Katz’s Pensari. Also, Robert Abbott has developed some Mad Mazes (1990, Bob Adams).
A simplified version of Eleusis, called Eleusis Express can be found at logicmazes.com.
Puzzle: What is Smith’s rule in the above game?
The Chief
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