Can You Balance The Target?

Riddler Express

From Kyle Willstatter comes a puzzle that’s right on target:

You’re playing darts and trying to maximize the number of points you earn with each throw. You are deciding which sector to aim for. Your dart has a 50 percent chance of landing in that sector and a 25 percent chance of landing in one of the two neighboring sectors. Reading clockwise, the sectors are worth 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12 and 5 points, as shown below. (For the purposes of this puzzle, don’t worry about the bullseye, the outer ring that’s worth double or the inner ring that’s worth triple.)

Which sector should you aim for to maximize your expected score?

Extra credit: How would you “fairly” (by some definition of fair for you to define) assign the point values around a dartboard? Explain your thinking.

Solution

The expected value of where one aims at is shown in green.

Answer

Aiming for 7 yields the highest expected value of 12.25.

Extra Credit

The probability of hitting each sector is $\dfrac{1}{20}$. Therefore, the expected value of any throw is
$$\dfrac{\sum 20}{20} = 10.5$$.
If each sector's expected value were 10.5, that would be the most "fairly" assigned point values.

However, that is impossible.

Given sector with value $v$, with neighbors $u$ and $w$:
$$0.25u + 0.5v + 0.25w = 10.5$$
$$u + 2v + w = 42$$.
Note that if $v = 1$, both neighbors would have to be 20. Conversely, if $v = 20$, both neighbors would have to be 1.

Since two consecutive sector values generate the third, I created the following dartboard, starting with 1 and 20.

Going clockwise, $10$ would need $12$ and $10$ as neighbors, which is impossible. Going counterclockwise, $11$ would need $9$ and $11$ as neighbors, which is also impossible.

The four sectors that have expected values $\ne 10.5$ have expected values as close as possible to $10.5$.

This seems to be objectively the most 'fairly' distributed assignment of numbers.

Rohan Lewis

2023.01.29

Code can be found here.