Triangle

Contributed by Patrick St-Louis

This 3-player game was inspired by induction puzzles such as the Hat Puzzle. It is played with a 24-card pack consisting of four each of the cards from 1 to 6 - this can be made by extracting all the aces, twos, threes, fours, fives and sixes from a standard pack, the ace having a value of 1 (the smallest value).

A round consists of the following steps:

1. One of the players takes the role of the "dealer" and deals one card to each player (face down) and one in the center (face down). The card in the center will be called the "Right to Speak".
2. The dealer counts down (aloud) from 3 to 0, at which point each player lifts his card to show it to the other players, without looking at it himself.
3. A player who wants to "state a fact" takes the "Right to Speak", looks at its value and then makes one of the statements listed below.
4. The three players' cards are exposed to all players and the player who stated a fact gains or loses points depending on whether the statement was correct. The "Right to Speak" card is not shown to the other players - only the player who stated a fact is allowed to know what it was.
5. The four used cards are set aside face down, and the cards for the next round will be dealt from the remaining unused cards.

At step 3, if two players reach for the "Right to Speak" at the same time such that it is unclear who got it first, the round is a draw: no one scores and no one sees his own card nor the "Right to Speak" card. The four cards are discarded and the next round is dealt.

A turn is a series of 6 rounds. At the end of a turn, all cards have been discarded. Simply reshuffle them to start a new turn.

Only the following facts can be stated. (Note: a card has an even value if it is a two, a four or a six, and it has an odd value if it is an Ace, a three or a five.)

1. "My peers are even" (meaning that the cards of the other two players are even)
2. "I am even" or "I am odd" (the player has deduced or guessed that his card has even or odd value)
3. "I am greater than my peers" (the player has deduced or guessed that his card has strictly higher value than those of the other two players)
4. "I am among my peers" (the player has deduced or guessed that his card has the same value as one or both of the cards of the other two players)
5. "I am an Ace/two/three/four/five/six" (the player states the exact value of his own card)

(Note that the statements "My peers are odd" and "I am less than my peers", though similar to 1 and 3 above, are not allowed).

Stating a fact correctly earns you points:

• Fact type 1: 1 point
• Fact type 2: 2 points
• Fact type 3: 3 points
• Fact type 4: 5 points
• Fact type 5: 8 points

But if you state a fact that isn't true, you score nothing and each of the other players scores 3 points.

The players keep playing turns until at the end of a turn one or more players has at least 25 points. The player(s) with most points then win(s).

Other formats are possible. For example the players may agree in advance on a different target score, or to play a specific number of turns. There can be degrees of winning and losing depending on the scores, where effectively each player pays or receives from each opponent the difference between their scores. The choice of formats will affect the straegy near the end of the game to some extent. However, it is important not to stop playing in the middle of a turn, because some players may use strategies that pay off later in a turn and it wouldn't be fair to them to stop too early.

Strategy

Basic strategy: if you see two even cards, take the "Right to Speak" and state "My peers are even" to earn 1 point. If you see one even card and one odd card, then wait a second or two. If no one took the "Right to Speak", then it means that your own card must be of odd value, otherwise someone would have taken the "Right to Speak" to state "My peers are even". So you take the "Right to Speak" and state "I am odd" to earn 2 points. If you see two odd cards, then wait longer - more than a second or two. If no one took the "Right to Speak", and you think they know and are following the basic strategy, then you can reason that if you had an even value, they would have taken the "Right to Speak" to declare themselves odd. So all three cards are odd and you take the "Right to Speak" and state "I am odd".

The whole game revolves around correctly interpreting what your peers are thinking, and how fast they do so. It has a direct impact on how long you will wait before trying to take the "Right to Speak". Also, you can deduce a lot by seeing who else tried to reach for it while you were taking it. All this information, along with your memory of which cards have been dealt so far, changes the probability of each fact being true or false. Hence, a genius could count the cards, calculate all the probabilities that each fact is right or wrong, and then state the fact that gives the greatest expected gain according to the probabilities.

Seeing two aces or an ace and a two or two twos makes it worthwhile to take the "Right to Speak" right away and state "I am greater than my peers", unless lots of higher valued cards have passed in previous rounds of course. Also, seeing an ace and a three or a two and a three makes it worth waiting only a second or two before taking the "Right to Speak", since at least one player sees an ace or a two and your card. This means that if you had an ace or a two, that player should have taken the "Right to Speak" right away (see previous point). So if that player doesn't do that, it might mean your card has value 3 or more. The odds that you are "greater than your peers" are good.

There are other induction tricks that can be applied when players gain some skill. For example, if you see a two and a three, wait a second or two and take the "Right to Speak". You can deduce that you do not have an even value. This means that you have an odd value. Now, if you had an ace, one player would have seen an ace and a two and taken the "Right to Speak" to try to earn 3 points (I am greater than my peers). So the odds are that you have an odd value and not an ace (i.e. a 3 or a 5). Guessing exactly for a 3 or a 5 for 8 points has better overall winning odds than any other fact (-3 + 8 divided by 2 equals 2.5). The odds are even better if you counted cards and know if there are more threes than fives left.