# Brain teaser

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A Brain teaser is a form of puzzle that involves a lot of thinking (mental/cognitive activity). Normally, this includes thinking in conventional ways with given constraints in mind; sometimes, it also involves lateral thinking. Logic puzzles and riddles are specific types of brain teasers.

Example:

Q: If three hens lay three eggs in three days, how many eggs does a hen lay in one day?
A: One third. (Note: 3 hens = 3 eggs / 3 days → 3 hens = (3 / 3) (eggs / days) → 1 hen = (1 / 3) (egg / days))

It is easy for people to argue about the answers of many brain teasers; in the given example with hens, one might claim that all the eggs in the question were laid in the first day, so the answer would be one, or comment that it is rare for a hen to lay a fraction of an egg.

The difficulty of many brain teasers relies on a certain degree of fallacy in human intuitiveness. This is most common in brain teasers relating to conditional probability, because the casual human mind tends to consider absolute probability instead. As a result, a great number of controversial discussions emerge from such problems, the most famous probably being the Monty Hall problem. Another (simpler) example of such a brain teaser is given here:

If we encounter someone with two children, given that at least one of them is a son, what is the probability that the other is also a son?

(Of course, for the purpose of simplicity, we will disregard hermaphrodites and assume that boys and girls are born with equal probability.) The common intuitive way of thinking is that the births of the two children are independent of each other, and so the answer must be the absolute probability of one child being a boy, 1/2. However, the correct answer is 1/3 as shown by the following argument:

• For a single birth, there are two possibilities (a boy or a girl) with equal probability.
• Therefore, for two births, there are four possibilities: 1) two boys, 2) two girls, 3) first a boy, then a girl, and 4) first a girl, then a boy; all of them have equal probability.
• We are given that one of the children is a boy. Thus, only one of the four possibilities -- two daughters -- is eliminated. Three possibilities with equal probabilities (1/3) remain.
• Out of those three, only one -- two sons -- is what we are looking for. Hence, the answer is 1/3.

Alternatively, one can see that in any sample of families with two children, 3/4 of them will have at least one son, and 1/4 will have two sons. The probability is thus (1/4)/(3/4) = 1/3. The common intuitive way of thinking is equivalent to considering families in which a particular child (e.g. the first-born, or the one that comes first in the alphabet, etc.) is a son (which is only 1/2 of the sample, not 3/4) and seeing how many of them have two sons.

One might formulate the above as

If someone has two children, and one of them is a son, what is the probability that the other is also a son?

but that would be (more) ambiguous, since it could mean that we chose a person at random, and learnt that at least one of their two children was a son (in which case we get 1/3), or it could mean that we chose a person at random, and met one of their children, which turned out to be a son. This would then be a particular child, so the probability of the other being a son is 1/2.