Last week I went to a talk by Jim Al-Kalili, called “9
Paradoxes” (which, not coincidentally, is the title of his latest book).

One style of paradox he talked about are the mathematical
ones, where a plausible calculation is presented, but the conclusion is
wrong. He illustrated this with the
Monty Hall problem, and the Missing Dollar puzzle.

I’ve written about the Monty Hall problem elsewhere, and how the resolution is much easier to see by taking a
more extreme case. The approach of
changing the problem to an extreme case is not specific to the Monty Hall
problem, but is a more widely applicable check.
In particular, is can be applied to help see through the Missing Dollar
puzzle.

The Missing Dollar puzzle is as follows:

Three friends book into a shared room at an hotel. The rate is $30, so they pay $10 each. Later, the clerk realises they have overpaid; the rate is actually $25. He takes $5 from the till, and goes to give them their refund. On the way he realises that he won’t be able to split $5 between the three, so gives them $1 each, and pockets the remaining $2.

So they have each paid $10-$1=$9, which is a total of $27. With the $2 in the clerk’s pocket, that’s a total of $29. The original payment was $30. What happened to the missing $1?

The answer is that this is the wrong calculation. They have
paid $27. Of this $2 is in the clerk’s
pocket, and $25 is in the till to pay for the room. The puzzle works because the two prices are
so close, and so it isn't necessarily obvious on a fast telling of the puzzle that the $2 should be subtracted from the $27, rather than added to it. Let’s use the same approach of
taking it to extremes to make the problem more obvious.

Three friends book into a shared room at an hotel. The rate is $3000, so they pay $1000 each. Later, the clerk realises they have overpaid; the rate is actually $25. He takes $2975 from the till, and goes to give them their refund. On the way he realises that he won’t be able to split $2975 between the three, so gives them $991 each, and pockets the remaining $2.

So they have each paid $1000-$991=$9, which is a total of $27. With the $2 in the clerk’s pocket, that’s a total of $29. The original payment was $3000. What happened to the missing $2971?

It is much clearer now that is that this is the wrong
calculation. They have paid $27. Of this
$2 is in the clerk’s pocket, and $25 is in the till to pay for the room.

Going to extremes doesn't work for everything, but it is quite a powerful argument sanity-checker.

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