The Missing Evidence

Two weeks ago when I introduced the puzzle of the two contestants trying to discover hidden digits (numbers, not toes), I mentioned that its charm was due to looking at what people don’t know rather than what they do know. In life as well as artificial puzzles, people often only look at positive evidence, and overlook the equally valuable negative, or missing evidence. [BTW, at least one reader solved it by making a spreadsheet of possibilities then following the consequences of each round of answers.]

My informal introduction to this concept was as a boy playing the game of Clue. For those of you who might never have played it, here is a quick synopsis. A murder has been committed somewhere in a nine-room mansion by one of six suspects using one of 6 possible weapons. At the start of the game, playing cards for rooms, suspects and weapons are shuffled separately. One card is taken from each pile and put in a holder. The object of the game is to discover who did it where and with what. The remaining cards are dealt to the players. If four people play, then the 18 remaining cards go out as (4, 4, 5, 5). Each player gets a notepad to keep track of clues. Life is not always fair, and the players with more cards start with more clues since the cards in a player’s hand are obviously not in the holder.

By various moves, players enter rooms and suggest how the foul deed was done (players must be in the suspected room when they make the suggestion). Then by rote the next player must prove the suggestion wrong if possible. This is done by secretly showing the person who suggested a scenario one of the cards that disproves it. If that player cannot disprove it, the next player must try, and so on. Play continues until someone makes a firm accusation. When an accusation is made, the accuser looks at the secret cards and wins if correct. False accusers are out of the game and probably never seen again in polite society.

Beginners simply cross off the cards they have seen while listening to the other suggestions for ideas of how to proceed. But with a little thought, one realizes that each time a player cannot disprove a suggestion, regardless of who proposed it, information is passed to all the other players about the contents of that person’s hand. Assume both you and the next person have 4 cards. Before you look at your hand, that opponent could have one of 5,985 possible hands. After you look at your hand, this number drops to 2,380. If that person cannot disprove a suggestion you make, then you know at least 7 cards that the person does not have, and the number of possible hands drops. If that same person cannot disprove another suggestion from any other player, with 3 different cards suggested, then you can eliminate another 3 cards as possibilities. This drops the pool of possible cards in your opponent’s hand to 11, and the number of possible hands to 330. The number of possible hands drops with each response, positive or negative. With some careful bookkeeping, one can quickly avoid making suggestions that would not yield information. Sometimes one can even decide what is in an opponent’s hand and therefore not in the holder.

Similarly, every time a person disproves a suggestion, even if you were not the one suggesting, you gain a clue because that person must have at least one the three cards. With careful bookkeeping and some thought, one can win without even going into the room where the murder occurred (getting from room to room can be a hassle and is best avoided if possible because it can take several turns to get the right combination of die tosses to make the trip), and that can confound your opponents.

By the time I figured this out and was winning all the games, my former friends started using other strategies like making suggestions when they had some or all of the cards. This can be a useful way to force disclosure of a given card or cause a distraction, but in the long run that strategy is counter-productive. Why?

After one summer, we abandoned Clue for Monopoly and Canasta. But the lesson I learned of looking at the missing data instead of depending on positive clues stayed with me and came in handy when playing poker in college, but that is another story.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

[tags]decision theory, strategy, puzzle, evidence, speculation[/tags]