You have a square and a random variable that picks a random point on the square with a uniform distribution. What is the probability that a randomly selected point is closer to the centre than to the edge? (Thanks to OMD.)

Solution

Many people will think that the required probability is the same as the probability of landing in the circle with diameter half the side of the square. But this is not the case. The line separating closer to centre from closer to edge is a parabola. The answer is

Snowflake

Start with an equilateral triangle. Now stick on to the middle of each side equilateral triangles with side one third of the side of the original triangle. This gives you a Star of David, with six points. Now add on to the sides of the six triangles yet smaller triangles, with side one third of the 'parent' triangle and so on ad infinitum. What are the perimeter and area of the final snowflake? (Thanks to Gerasimos.)

Solution

First count how many sides there are as a function of number of iterations. Initially there are three sides, and then 3 x 4. Every iteration one side turns into four. So there will be 3.4n after n iterations. The length of each side is one third what it was originally. Therefore after n iterations the perimeter ,A,ill KQ

multiplied by the original perimeter. It is unbounded as n tends to infinity.

The area increases by one third after the first iteration. After the second iteration you add an area that is number of sides multiplied by area of a single small triangle which is one ninth of the previously added triangle. If we use An to be the area after n iterations (when multiplied by the area of initial triangle) then

The final calculation exploits the binomial expansion.

This is the famous Koch snowflake, first described in 1904, and is an example of a fractal.

The doors

There are 100 closed doors in a corridor. The first person who walks along the corridor opens all of the doors. The second person changes the current state of every second door starting from the second door by opening closed doors and closing open doors. The third person who comes along changes the current state of every third door starting from the third door. This continues until the 100th person. At the end how many doors are closed and how many open? (Thanks to zilch.)

Solution

This is a question about how many divisors a number has. For example the 15th door is divisible by 1, 3, 5 and 15. So it will be opened, closed, opened, closed. Ending up closed.

What about door 37? Thirty-seven is only divisible by 1 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 1 to 100. Of course, there are only 10.

Two thirds of the average

Everyone in a group pays $1 to enter the following competition. Each person has to write down secretly on a piece of paper a number from 0 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 the Financial 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 winning 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 0 because if everyone submits the answer 0 then two thirds of that is still 0, and so 0 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 that 0 is 'correct' or they stop the inductive process after a couple of iterations and submit something around 20. It may be that the longer people have to think about this, the lower the number they submit.

This is a nice problem because it does divide people into the purely rational, game-theoretic types, who pick 0, and never win, and the more relaxed types, who just pick a number after a tiny bit of thought and do stand a chance of winning.

Personal note from the author: The Financial Times ran this as a competition for their readers a while back. (The prize was a flight in Concorde, so that dates it a bit. And the cost of entry was just the stamp on a postcard.)

I organized a group of students to enter this competition, all submitting the number 99 as their answer (it wasn't clear from the rules whether 100 was included). A number which could obviously never win. The purpose of this was twofold, (a) to get a mention in the paper when the answer was revealed (we succeeded) and (b) to move the market (we succeeded in that as well).

There were not that many entries (about 1,500 if I remember rightly) and so we were able to move the market up by one point. The FT printed the distribution of entries, a nice exponentially decaying curve with a noticeable 'blip' at one end! The winner submitted the number 13.

I didn't tell my students this, but I can now reveal that I secretly submitted my own answer, with the purpose of winning... my submission was 12. Doh!

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