# Ages of three children

A census taker goes to a house, a woman answers the door and says she has three children. The census taker asks their ages and she says that if you multiply their ages, the result is 36. He says he needs more info so she tells him that the total of their ages is the address of the building next door. He goes and looks, then comes back and says he still needs more information. She tells him that she won t answer any more questions because her eldest child is sleeping upstairs and she doesn't want to wake him.

What are the children s ages? (Thanks to tristanreid.)

**Solution**

First suitably factorize 36: (1,1,36), (1,4,9), (1,2,18), (1,3,12), (1,6,6), (2,3,6), (2,2,9), (3,3,4).

When the census taker is unable to decide from the information about nextdoor s house number we know that nextdoor must be number 13, because both (1,6,6) and (2,2,9) add up to 13. All of the other combinations give distinct sums. Finally the mother refers to the 'eldest child, and this rules out (1,6,6) because the two older children have the same age. Conclusion the ages must be 2, 2 and 9.

Caveat: (1,6,6) is technically still possible because one of the six-year olds could be nearing seven while the other has only just turned six.

# The Monty Hall problem

You are a contestant on a gameshow, and you have to choose one of three doors. Behind one door is a car, behind the others, goats. You pick a door, number 2, say, and the host, who knows what is behind each door, opens one of the other two doors, number 3, say, and reveals a goat. He then says to you, ''Do you want to change your mind, and pick door number 1?''

Should you?

**Solution**

This is a classic question and is based on the real-life American gameshow called *Let's Make a Deal.*

Assuming that you prefer cars to goats then the correct thing to do is change door. (There is a twist though, to be explained at the end.) However, as you will probably know if you ever watch a magic show, people are more often than not reluctant to change their minds for reasons to do with belief in fate, and possible regret. (If you choose the wrong door and don't change then that was fate, it just wasn't your lucky day. If you choose correctly and then change then it is your 'fault. )

Some people think the answer to this question is counterintuitive. I don't. But let's do the maths anyway.

Suppose you don t change door. The probability of you having already picked the correct door remains at one in three. It s easy to see this, just imagine that you didn't hear the gameshow host, or you had your eyes closed when being given the option to change.

That leaves a probability that if you change to the only other possible remaining door, the probability is two thirds. Therefore change.

You can also argue as follows, assuming the car is behind door 1:

• You pick door 1. Host opens one of the other doors, it doesn't matter which. You change. You lose.

• You pick door 2. Host must open door 3, because the car is behind door 1! You change, and win.

• You pick door 3. Host must open door 2, because the car is behind door 1. You change, and win.

Hence the odds.

And now the twist. Imagine the following scenario. You are going for a quant interview. You are doing well, and the interviewer casually says 'Have you heard of the Monty Hall problem? You reply that you have. The interviewer picks up three paper cups and a coin. He asks you to turn away while he puts the coin under one of the cups. 'Ok, you can look now. We are going to play a little game. If you can find the coin I will give you a job. Fantastic, you think, knowing Monty Hall you are confident of a two thirds chance of getting the job!

'Pick a cup.' You pick cup number 1.

What happens next is subtle.

The interviewer lifts up cup number 2 to reveal nothing. 'Would you like to change your mind? Of course, you would! So you say yes, the interviewer lifts up cup number 3 to reveal ... nothing! You leave unemployed.

That was one scenario. It happens one third of the time. There is another scenario though. Go back to when you picked cup number 1. And without any comment, the interviewer picks up that cup, and beneath it, nothing! That happens two thirds of the time, and again no job.

Do you see what happened? The interviewer was only playing Monty Hall when it was to his advantage to do so. When you pick the wrong cup initially, as you will more often than not, then it s not Monty Hall! After all, the interviewer didn't say you were going to play the Monty Hall game, he only asked if you d heard of it. Subtle, but it cost you the job.

This is related to a true story of a friend of mine, a very clever person, going for a job to work for a household-name quant. My friend was asked about Monty Hall by a quant who only knew the basics of the problem, but my friend knew about the game-theory aspects. An argument ensued and my friend did not get the job, despite clearly being brighter than the interviewer!