Chapter 10: Brainteasers 375
opened, closed. Ending up closed. What about door 37?
Thirty seven is only divisible by one and 37. But again
it will end up closed. Since only squares have an odd
number of divisors we have to count how many squares
there are from one to 100. Of course, there are only 10.
Two thirds of the average
Everyone in a group pays $1 to enter the following com-
petition. Each person has to write down secretly on a
piece of paper a number from zero to 100 inclusive.
Calculate the average of all of these numbers and then
take two thirds. The winner, who gets all of the entrance
fees, is the person who gets closest to this final number.
The players know the rule for determining the winner,
and they are not allowed to communicate with each
other. What number should you submit?
(Thanks to knowtorious and theFinancial Times.)
Solution
This is a famous economics experiment, which examines
people’s rationality among other things.
If everyone submits the number 50, say, then the win-
ning number would be two thirds of 50, so 33. Perhaps
one should therefore submit 33. But if everyone does
that the winning number will be 22. Ok, so submit that
number. But if everyone does that...You can see where
this leads. The stable point is clearly zero because if
everyone submits the answer zero then two thirds of
that is still zero, and so zero is the winning number.
The winnings get divided among everyone and there
was no point in entering in the first place.
In discussions about this problem, people tend to carry
through the above argument and either quickly conclude