Answer: Tuesday Teaser #32
It is clear from the format of the game that by playing entirely by random, the chance of winning is 1/3. For example, an entirely random strategy could mean simply choosing a random piece of paper and sticking with it, whatever it may be. But we can do a lot better than this by improving our strategy.
A helpful concept when trying to come up with optimal strategies is the concept of information. If we can obtain more information, we can make a more informed decision that will give us a better chance of winning.
In this game, that information is how the numbers on the pieces of paper are related to each other. In order to gain such information, it is clear that we have to turn over at least two pieces of paper, which means that we have to discard the first piece of paper, whatever number we uncover.
Having turned over the second piece of paper, we now have a simple either-or choice; we can stick with the piece we’ve got, or discard it and gamble on the third piece.
It is clear that if the second number is lower than the first, then it can’t be the highest, so we should always discard it and gamble on the third number.
But if the second number is higher than the first, what should we do? Well, there are three situations in which this happens. In two of the three cases, the second number will actually be the higher number (see the full list of possibilities below), so we should stick, rather than twist, to maximise our chances of choosing it.
This strategy ends up with a 1/2 chance of being successful; much better than the 1/3 random chance. The reason for this increase is that using the better strategy we obtained extra information and made a choice based on that extra information.
Below is the full list of possibilities of the order of the three pieces of paper, where 1 denotes the highest number and 3 the lowest. The number in bold is the one chosen by the optimal strategy, so note that we win 3 out of 6 times.
- 1 => 2 => 3
- 1 => 3 => 2
- 2 => 1 => 3
- 2 => 3 => 1
- 3 => 1 => 2
- 3 => 2 => 1
