# Entangled physicists

The anti-intellectuals have won the war, and under the iron rule of emperor Nonotin anyone who has been deemed too smart for their own good has been brought to labour camps.

But most of the science folks turn out to have never done a fair day's work in their life, so they are worth little in the labour camps. In order to get rid of some of them, and to have a little laugh, Nonotin devises an impossible game based on an equally impossible heretic tale. Though he doesn't care much for the exact details of the tale.

He places 1 cat in each of 20 lidded boxes, and asks one of his guards to kill a random number of them. He then gathers 100 of the most useless physicists, explain this setting to them, and states the rules of the game.

The physicists will, one at a time, be taken to the room with the boxes, look into 19 of them, and then guess if the cat in the last one is living or dead. The boxes will be moved around between each visit, so the physicists will not be able to coordinate their choice of box.

If they all guess correctly they will be given their freedom, but if anyone guesses wrong they will all be executed.

The physicists only have a short while to agree on a strategy before being taken to isolation cells. They don't think it likely that they will reliably be able to hear the living cats, or otherwise identify the content of the boxes without looking.

The emperor is certain of their defeat. But is there truly nothing the physicists can do to have a fair chance of winning the game?

• Surely if they look in 19 boxes it will be obvious whether the remaining cat is alive or dead. And I'm really curious what "tentaclement" is, and how I can use it myself. – Kendall Frey Sep 6 '14 at 22:24
• @KendallFrey No one told them how many cats the guard killed. – aaaaaaaaaaaa Sep 6 '14 at 22:28
• Are the physicians able to kill the cats they observe before making their guess? If so, first one leaves 19 corpses and guesses "dead" for the last one - if wrong, all is lost anyway; if right, the other physicians know all cats are dead before entering the room... – Julia Hayward Sep 8 '14 at 9:36
• As long as nobody opens up the box to check on the cat, they can all guess its a superimposed wave state of living death. 100% chance of success. – DiscOH Sep 8 '14 at 18:49
• Since this is based on Schrödinger's cat, I think you mean physicists rather than physicians (medics). – Peter Taylor Sep 9 '14 at 8:28

## 2 Answers

Assuming that "kill a random number of kittens" means a random integer from 0 to 20, then there is a 11/21 chance that an even number of kittens have been killed.

The physicians should all agree to guess that the last kitten is dead if an odd number of kittens are dead, or alive if an even number of kittens are dead.

That's an approximately 52% chance of staying alive, but a less than 5% chance that no kittens were harmed in the making of this puzzle.

• It is a cruel world and a cruel puzzle. It is worth noting that it doesn't matter much how the guard interpret his order, they'll end up with approximately 50% survival chance if only he doesn't favour killing an odd number of cats. – aaaaaaaaaaaa Sep 7 '14 at 7:31
• So this is an improvement of 2 percentage points over just agreeing that the last cat is dead? – justhalf Sep 8 '14 at 5:14
• @justhalf No. The boxes are moved around, so the chances that two physicists have the same last cat is 1/20. That makes it near certain that just guessing 'dead' will result in dead physicists. – Joel Rondeau Sep 8 '14 at 12:41
• It doesn't have to point to the same box. The chance that a box contain a dead cat is 0.5 (average of 0/20, 1/20, 2/20, ..., 20/20). Hmm, but yes, I think there is something wrong with this argument. This way the chance is 0.5^n, where n is the number of physichists. – justhalf Sep 8 '14 at 12:46
• But @TimCouwelier, the idea here is that there isn't a 52% chance for each individual, but overall. That is, there is a 52% chance that there are an even number of cats. Since everyone is guessing that same event, and giving an answer that is logically equivalent, either everyone is right or everyone is wrong. So it's really (52 / 0.50^100), a huge improvement. – TheRubberDuck Sep 9 '14 at 15:34

They should do the exact opposite of what Joel Rondeau suggests, and gamble on the number of dead cats being odd.

If the guard were an intellectual then he might use a good randomisation technique. But he's been selected for his anti-intellectualism, so he will pick a "random" number mentally. When humans pick "random" numbers like that, we have a definite bias towards odd numbers.

So if the intellectuals assume that the guard will kill an odd number of cats, they probably have about a 60% chance of survival.

• Yes. As the way to choose randomness was not defined in the puzzle, this is almost certainly a better choice of real life "random" than mine was. – Joel Rondeau Sep 9 '14 at 13:10
• There's also a sort of bias that certain numbers can't be generated randomly... So maybe the dumb guard would think "I can't kill them all, that wouldn't be random!". The same might apply for 0 and 10. – TheRubberDuck Sep 9 '14 at 15:38
• If they just try to kill each cat, and they succeed with 50% chance, then it doesn't matter whether you choose even or odd. – Florian F Sep 14 '14 at 11:43
• Another thing to consider is that a dumb guard might decide that flipping a coin for each cat is random. But he doesn't have a coin so he determines that exactly half is "random." Half is an even number :( – acbabis Oct 6 '14 at 17:32
• @FlorianF If you, as a guard, fail 50% of the time at killing a mere cat, you should be fired for incompetence. – Nzall Oct 21 '14 at 9:41