From: flee@cs.psu.edu (Felix Lee)
Subject: Re: REQUEST: Eleusis
Date: Fri, 20 Nov 1992 07:09:19 GMT

[Okay.  Here's part of the New Eleusis chapter from Martin Gardner's
_Penrose Tiles to Trapdoor Ciphers_.]

[...]

Eleusis was invented in 1956 by Robert Abbott of New York, who at the
time was an undergraduate at the University of Colorado.  He had been
studying that sudden insight into the solution of a problem that
psychologists sometimes call the "Aha" reaction.  Great turning points
in science often hinge on these mysterious intuitive leaps.  Eleusis
turned out to be a fascinating simulation of this facet of science,
even though Abbott did not invent it with this in mind.  In 1963
Abott's complete rules for the game appeared in his book, _Abbott's
New Card Games_ (hardcover, Stein & Day; paperback, Funk & Wagnalls).

Martin D. Kruskal, a distinguished mathematical physicist at Princeton
University, became interested in the game and made several important
improvements.  In 1962 he published his rules in a monograph titled
_Delphi: A Game of Inductive Reasoning_.  Many college professors
around the country used Eleusis and Delphi to explain scientific
method to students and to model the Aha process.  Artificial
intelligence scientists wrote computer programs for the game.  at the
System Development Corporation in Santa Monica, research was done on
Eleusis under the direction of J. Robert Newman.  Litton Industries
based a full-page advertisement on Eleusis.  Descriptions of the game
appeared in European books and periodicals.  Abbott began receiving
letters from all over the world with suggestions on how to make
Eleusis a more playable game.

[...]

At least four players are required.  As many as eight can play, but
beyond that the game becomes too long and chaotic.

Two standard decks, shuffled together are used.  (Occasionally a round
will continue long enough to require a third deck.)  A full game
consists of one or more rounds (hands of play) with a different player
dealing each round.  The dealer may be called by such titles as God,
Nature, Tao, Brahma, the Oracle (as in Delphi) or just Dealer.

The dealer's first task is to make up a "secret rule".  This is simply
a rule that defines what cards can be legally played during a player's
turn.  In order to do well, players must figure out what the rule is.
The faster a player discovers the rule, the higher his score will be.

One of the cleverest features of Eleusis is the scoring (described
below), which makes it advantageous to the dealer to invent a rule
that is neither too easy to guess nor too hard.  Without this feature
dealers would be tempted to formulate such complex rules that no one
would guess them, and the game would become dull and frustrating.

An example of a rule that is too simple is: "Play a card of a color
different from the color of the last card played."  The alternation of
colors would be immediately obvious.  A better rule is: "Play so that
primes and nonprimes alternate."  For mathematicians, however, this
might be too simple.  For anyone else it might be too difficult.  An
example of a rule that is too complicated is: "Multiple the values of
the last 3 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
1, play a black card or a picture card.  If the remainder is 2, play
an even card or a card with a value lower than 6.  If the remainder is
3, play an odd card or a 10."  No one will guess such a rule, and the
dealer's score will be low.

Here are three examples of good rules for games 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 the last card played was red, play a card of equal
or lower value.  (The values of the jack, queen, king and ace are
respectively 11, 12, 13 and 1.)

3. The card played must be either of the same suit or the same value
as the last card legally played.

The secret rules must deal only with the sequence of legally played
cards.  Of course, advanced players may use rules that refer to the
entire pattern of legal and illegal cards on the table, but such rules
are much harder to guess and are not allowed in standard play.  Under
no circumstances should the secret rule depend on circumstances
external to the cards.  Examples of such improper rules are those that
depend on the sex of the last player, the time of day, whether God
scratches his (or her) ear and so on.

The secret rule must be written down in unambiguous language, on a
sheet of paper that is put aside for future confirmation.  As Kruskal
proposed, the dealer may give a truthful hint before the play begins.
For example, he may say "Suits are irrelevent to the rule", or "The
rule depends on the two previously played cards".

After the secret rule has been recorded, the dealer shuffles the
double deck and deals 14 cards to each player and none to himself.  He
places a single card called the "starter" at the extreme left of the
playing surface, as is indicated in [figure].  To determine who plays
first the dealer counts clockwise around the circle of players,
starting with the player on his left and excluding himself.  He counts
until he reaches the number on the starter card.  The player indicated
at that number begins the play that then continues clockwise around
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 his hand and shows it to
everyone.  If according to the rule the card is playable, the dealer
says "Right".  The card is then placed to the right of the starter
card, on the "main line" of correctly played cards extending
horizontally to the right.

If the card fails to meet the rule, the dealer says "Wrong".  In this
case the card is placed directly below the last card played.  Vertical
columns of incorrect cards are called "sidelines".  (Kruskal
introduced both the layout and the terminology of the main line and
sidelines.)  Thus consecutive incorrect plays extend the same sideline
downward.  If a player displays a wrong card, the dealer gives him two
more cards as a penalty, thereby increasing his hand.

If a player thinks he has discovered the secret rule, he may play a
"string" of 2, 3 or 4 cards at once.  To play a string he overlaps the
cards slightly to preserve their order and shows them to everyone.  If
all the cards in the string conform to the rule, the dealer says
"Right".  Then all the cards are placed on the main line with no
overlapping, as if they were correctly played single cards.

If one or more cards in a string are wrong, the dealer declares the
entire string wrong.  He does not indicate which cards do not conform
to the rule.  The wrong cards are left overlapping to keep their
identity as a string and the entire string goes below the last card
played.  The player is then dealt twice as many cards as there are in
the string.

[...]

Players improve their score by getting rid of as many cards as
possible, and of course they can do this best if they guess the secret
rule.  At the start of a round there is little information to go on,
and plays are necessarily random.  As the round continues and more and
more information is added to the layout, the rule becomes steadily
easier to guess.

It may happen that a player thinks he knows the secret rule but finds
he has no card that can be legally played.  He then has the option of
declaring "No play".  In this case he shows his hand to everyone.  if
the dealer declares him right and his hand contains four cards or
less, the cards are returned to the deck and the round ends.  If he is
right and has five or more cards, then his cards are put back into the
deck, and he is dealt a fresh hand with four fewer cards than he
previously held.

If the player is wrong in declaring no play, the dealer takes one of
his correct cards and puts it on the main line.  The player keeps the
rest of his hand and, as a penalty, is dealt five more cards.  A
player who thinks he has no correct play but has not figured out the
secret rule should realize that the odds are against his using the no
play option successfully.  He would do better to play a card at
random.

When a player thinks he knows the secret rule, he has the opportunity
to prove it and increase his score.  He does so by delcaring himself a
Prophet.  The Prophet immediately takes over the dealer's duties,
calling plays right or wrong and dealing penalty cards when the others
play.  He can declare himself a Prophet only if all the following
conditions prevail:

1. He has just played (correctly or incorrectly), and the next player
has not played.

2. There is not already a Prophet.

3. At least two other players beside himself and the dealer are still
in the round.

4. He has not been a Prophet before in this round.

When a player declares himself a Prophet, he puts a marker on the last
card he played.  A chess king or queen may be used.  The Prophet keeps
his hand but plays no more cards unless he is overthrown.  The play
continues to pass clockwise around the players' circle, skipping the
Prophet.

Each time a player plays a card or string, the Prophet calls the play
right or wrong.  The dealer then either validates or invalidates the
Prophet's statement by saying "Correct" or "Incorrect".  If the
Prophet is correct, the card or string is placed on the layout---on
the main line if right or on a sideline if wrong---and the Prophet
gives the player whatever penalty cards are required.

If the dealer says "Incorrect", the Prophet is instantly overthrown.
He is declared a False Prophet.  The dealer removes the False
Prophet's marker and give him five cards to add to his hand.  He is
not allowed to become a Prophet again during the same round, although
any other player may do so.  The religious symbolism is obvious, but
as Abbott points out, there is also an amusing analogy here with
science: "The Prophet is the scientist who publishes.  The False
Prophet is the scientist who publishes too early."  It is the fun of
becoming a Prophet and of overthrowing a False Prophet that is the
most exciting feature of New Eleusis.

After a Prophet's downfall the dealer takes over his former duties.
He completes the play that overthrew the Prophet, placing the card or
string in its proper place on the layout.  If the play is wrong,
however, no penalty cards are given.  The purpose of this exemption is
to encourage players to make unusual plays---even deliberately wrong
ones---in the hope of overthrowing the Prophet.  In Karl Popper's
language, it encourages scientists to think of ways of "falsifying" a
colleague's doubtful theory.

If there is a Prophet and a player believes he has no card to play,
things get a bit complicated.  This seldom happens, and so you can
skip this part of the rules now and refer to it only when the need
arises.  There are four possibilities once the player declares no
play:

1. Prophet says, "Right"; dealer says "Correct".  The Prophet simply
follows the procedure described earlier.

2. Prophet says, "Right"; dealer says, "Incorect".  The Prophet is
immediately overthrown.  The dealer takes over and handles everything
as usual, except that the player is not given any penalty cards.

3. Prophet says, "Wrong"; dealer says, "Incorrect".  In other words,
the player is right.  The Prophet is overthrown, and the dealer
handles the play as usual.

4. Prophet says, "Wrong"; dealer says, "Correct".  In this case the
Prophet now must pick one correct card from the player's hand and put
it on the main line.  If he does this correctly, he deals the player
the five penalty cards and the game goes on.  It is possible, however,
for the Prophet to make a mistake at this point and pick an incorrect
card.  If that happens, the Prophet is overthrown.  The wrong card
goes back into the player's hand and the dealer takes over with the
usual procedure, except that the player is not given penalty cards.

After 30 cards have been played and there is no Prophet in the game,
players are expelled from the round when they make an incorrect play,
that is, if they lay a wrong card or make a wrong declaration of no
play.  An expelled player is given the usual penalty cards for his
final play and then drops out of the orund, retaining his hand for
scoring.

If there is a Prophet, expulsions are delayed until at least 20 cards
have been laid down after the Prophet's marker.  Chess pawns are used
as markers so that it is obvious when expulsion is possible.  As long
as there is no Prophet, a white pawn goes on every 10th card placed on
teh layout.  If there is a Prophet, a black pawn goes on every 10th
card laid down after the Prophet's marker.  when a Prophet is
overthrown, the black pawns and the Prophet's marker are removed.

A round can therefore go in and out of the phase when expulsions are
possible.  For example, if there are 35 cards on the layout and no
Prophet, Smith is expelled when he plays incorrectly.  Next Jones
plays correctly and declares herself a Prophet.  If Brown then plays
incorrectly, she is not expelled because 20 cards have not yet been
laid down after the Prophet's marker.

A round can end in two ways: (1) when a player runs out of cards or
(2) when all players (excluding a Prophet, if there is one) have been
expelled.

The scoring in Eleusis is as follows:

1. The greatest number of cards held by anyone (including the Prophet)
is called the "high count".  Each player (including the Prophet)
subtracts the number of cards in his hand from the high count.  The
difference is his score.  If he has no cards, he gets a bonus of four
points.

2. The Prophet, if there is one, also gets a bonus.  It is the number
of main-line cards that follow his marker plus twice the number of
sideline cards that follow his marker, that is, a point for each
correct card since he became a Prophet and two points for each wrong
card.

3. The dealer's score equals the highest score of any player.  There
is one exception: If there is a Prophet, count the number of cards
(right and wrong) that precede the Prophet's marker and double this
number; if the result is smaller than the highest score, the dealer's
score is that smaller number.  [This discourages collaboration between
the dealer and a Prophet.]

If there is time for another round, a new dealer is chosen.  In
principle the game ends after every player has been dealer, but this
could take most of a day.  To end the game before everyone has dealt,
each player adds his scores for all the rounds played plus 10 more
points if he has not been a dealer.  This compensates for the fact
that dealers tend to have higher-than-average scores.

Some miscellaneous advice from Abbott should help inexperienced
Eleusis players.  Since layouts tend to be large, the best way to play
the game is on the floor.  Of course a large table can be used as well
as miniature cards on a smaller table.  If necessary, the main line
can be broken on the right and continued below on the left.

Remember that in Eleusis the dealer maximizes his score by choosing a
rule that is neither too easy nor too difficult.  Naturally this
depends both on how shrewdly the dealer estimates the ability of the
players and how accurately he evaluates the complexity of his rule.
Both estimates require considerable experience.  Beginning players
tend to underestimate the complexity of their rules.

For example, the rule used in the first layout is simple.  Compare it
with: "Play a red card, then a black card, then an odd card, then an
even card and repeat cyclically."  This rule seems to be simpler, but
in practice the shift from the red-black variable to the even-odd
variable makes it difficult to discover.  Abbott points out that in
general restrictive rules that allow only about a fourth of the cards
to be acceptable on any given play are easier to guess than less
restrictive rules that allow half or more of the cards to be
acceptable.
--
Newsgroups: rec.games.abstract