Games - Induction Puzzles

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	Games - Induction Puzzles

Induction Puzzles are Logic puzzles which are solved via the application of the principle of induction. In most cases, the puzzle's scenario will involve several participants with reasoning capability (typically people) and the solution to the puzzle will be based on identifying what would happen in an obvious case, and then repeating the reasoning that: "as soon as one of the participants realises that the obvious case has not happened, they can eliminate it from their reasoning, so creating a new obvious case".

Typical tell-tale features of these puzzles include any puzzle in which each participant has a given piece of information about all other participants but not themselves; also, usually some kind of hint is given to suggest that the participants can trust each others intelligence.

Examples

The King's Wise Men: The King called the three wisest men in the country to his court to decide who would become his new advisor. He placed a hat on each of their heads, such that each wise man could see all of the other hats, but none of them could see their own. Each hat was either white or blue. The king gave his word to the wise men that at least one of them was wearing a blue hat - in other words, there could be zero, one, or two white hats, but not three. The king also announced that the contest would be fair to all three men. The wise men were also forbidden to speak to each other. The king declared that whichever man stood up first and annouced the colour of his own hat would become his new advisor. The wise men sat for a very long time before one stood up and correctly announced the answer. What did he say, and how did he work it out?

Josephine's Problem: In Josephine's Kingdom every woman has to take a logic exam before being allowed to marry. Every married woman knows about the fidelity of every man in the Kingdom except for her own husband, and etiquette demands that no woman should tell another about the fidelity of her husband. A gunshot fired in any house in the Kingdom will be heard in any other. Queen Josephine announced that unfaithful men had been discovered in the Kingdom, and that any woman knowing her husband to be unfaithful was required to shoot him at midnight following the day she discovered his infidelity. How did the wives manage this?

Alice at the Convention of Logicians: At the Secret Convention of Logicians, the Master Logician placed a band on each attendee's head, such that everyone else could see it but the person themselves could not. There were many, many different colours of band. The Logicians all sat in a circle, and the Master instructed them that a bell was to be rung in the forest at regular intervals: at the moment when a Logician knew the colour on his own forehead, he was to leave at the next bell. Anyone who left at the wrong bell was clearly not a true Logician but an evil infiltrator and would be thrown out of the Convention post haste; but the Master reassures the group by stating that the puzzle would not be impossible for anybody present. How did they do it?

Solutions

The King's Wise Men: This is one of the simplest induction puzzles and one of the clearest indicators to the method used.

Suppose that you are one of the wise men. Looking at the other wise men, you see they are both wearing white hats. Since the king specified that there were at most two white hats, you would immediately know that your own hat must be blue.
Now suppose that you see the other wise men, and one is wearing a white hat and the other is wearing a blue hat. If your own hat was white, then the man you can see wearing the blue hat would be himself seeing two white hats and would - by the logic above - have immediately declared his hat colour. If he doesn't do this, it can only be because your hat isn't white, therefore it must be blue.
Now suppose that you see the other wise men and both are wearing blue hats. You can't work anything out from this. However, if your own hat was white, then one of the two other wise men would be seeing a blue and a white hat, and would have declared his hat colour by the rule above. Thus, if he hasn't done so, he must also be seeing two blue hats and thus your hat must be blue.

Note also that, in all the rules above, it is only possible for a person wearing a blue hat to win. Thus, since the king said that the contest would be fair to all three men, that must be the configuration in use - and the wise man announced that his hat was blue.

Josephine's Problem: This is another good example of a general case.

If there is only 1 unfaithful husband, then every woman in the Kingdom knows that except for his wife, who believes that everyone is faithful. Thus, as soon as she hears from the Queen that unfaithful men exist, she knows her husband must be unfaithful, and shoots him.

[ Visit the complete Wikipedia entry for Induction Puzzles ]

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