# How to effectively hide an earth rebooter thingy [The Good Place Season 4 spoilers]

This puzzle is unsolved. I don't know if there is a solution or how to solve it. I'm also not sure what type of puzzle this would be, so if anyone wants to correct the tags, feel free.

Imagine you have 38 Janets, one of whom is hiding a device which will reboot the Earth and all of humanity. The Judge is trying to find said device by searching each Janet one at a time, which might take her a minute or more. After her search, the Judge will kill the Janet if the Janet doesn't have the earth-rebooter-thingy.

Assume that:

1. The Janets are arranged in a line from left to right.

2. The Judge can start anywhere in the line and go in either direction, but once she starts in that direction, she can't change direction until she gets to the end of the line and has to start again.

3. The Janets don't know where or from which Janet the Judge is going to start searching.

4. The Judge finding the device is inevitable.

5. The Janets want to delay finding the device for as long as possible.

The question is, how would you decide beforehand which Janet to give the earth-rebooter-thingy to maximise the time it takes to find? Is this a solvable problem?

This is only solvable in the sense that no solution is better than another.

Since the judge has no prior knowledge, starting at any position is just as good, so the judge might as well toss a (38-sided) coin to decide, (treating the line as if its ends were connected to each other) and this is still optimal for the judge.

If the judge does that, it becomes clear that the position of the rebooter-Janet is irrelevant; the coin always has an equal chance of rearranging the device to any position.

• I had a feeling that would be the case. Do you think that an optimal solution for the Janets would arise if the Judge was only able to start at one end of the line, from left to right or from right to left? Then it would be a question of which of a 50/50 set of options do the Janets think the Judge will take.
– Lou
Jan 11 '20 at 15:59
• @Lou The judge can always assume that the Janets will correctly predict all his intentions and reasonings. This is a common game theoretical situation; it sometimes applies to stock market trading and other similar real world situations too. In such a situation, it is best to choose randomly, so that the adversary cannot get any advantage from predicting your choices.
– Bass
Jan 11 '20 at 16:07

Let's assume for the moment that the judge chooses the starting location at random uniformly, and similarly the direction of travel.

In that case, the location at either end of the row is the best for the Janets. Clearly every one of them is equally likely to be the first to be examined. The ones at the ends have only one neighbour, while all the rest have two. They are therefore less likely to be the second to be examined compared to the others. Similarly, when looking at the third, fourth, etc to be examined, the ones towards the middle are more likely.

The judge may well be able to partially compensate for this by choosing his starting location in a non-uniform way, being a bit more biased towards the outer ends of the row. I don't know if the judge can do it in such a way that everything becomes perfectly evenly balanced.

• When the judge reaches the end of the line, it's possible to continue from the far end of the remaining line, in the same direction. This essentially turns the line into a circle, with no bias whatsoever. This makes the worst case scenario identical to the best case scenario, rendering the Janets' decision irrelevant.
– Bass
Jan 11 '20 at 18:30
• @Bass You're right. I didn't really think about using a second round with a fixed strategy. Jan 11 '20 at 18:50