A quaint little puzzle...

One morning Mr Black, Mr Grey and Mr White decide to resolve a conflict by truelling with pistols until only one of them survives. Mr Black is the worst shot, hitting his target on average only one time in three. Mr Grey is a better shot hitting his target two times out of three. Mr White is the best shot hitting his target every time.

To make the truel fairer Mr Black is allowed to shoot first, followed by Mr Grey (if he is still alive), followed by Mr White (if he is still alive), and round again until only one of them is alive.

The question is this: Where should Mr Black aim his first shot?

[This was taken from a book which I will reveal once the answer has been given. You must give reasons for your answer because there are only a limited amount of answers]

Come on, someone have a stab at this one! I’ll reveal the answer if no-one answers soon…

i agree with erkyle

Some nice thinking there but Mr Black has an even better option.

The clue was in the question: “Where should he aim?” rather than “Who should he shoot?”

Mr. Black should of course shoot up into the sky (i.e. no-one). Next steps up Mr. Grey who will shoot for Mr.White because he is the biggest threat. If he misses, Mr White will definitely shoot Mr. Grey and if he shoots Mr.White then Mr.White is dead. Either, Mr. Black has one less opponent to deal with increasing his odds of survival. clever eh?

Originally posted by ben:

Mr. Black should of course shoot up into the sky (i.e. no-one.)

ok so he does this, but with his bad shooting he hits Mr. Grey. This leaves Mr. White with only one option: to shoot down Mr. Black!!

Interesting puzzle, Ben. You mentioned that you got it from a book: what book was that? I’d be interested in having a look. I’ve never really found a good book of puzzles like this, so if you know of any others I’d appreciate it if you list them here. Cheers.

Funnily enough the puzzle comes from “Fermat’s Last Theorem” by Simon Singh which gives a chronological account of Fermat’s Last Theorem, how it came about and how it was proved. The puzzle is common in game theory so no doubt you will find it elsewhere. The book itself is interesting for anyone who has an interest in maths or the history of maths. It doesn’t require any mathematical knowledge and it doesn’t even begin to try and explain the proof of fermat’s last theorem since only a handful of people alive can understand it!

This is like the whole situation with Zeno and the tortoise. Hopefully some of you have heard it because am suffering from an almighty hangover so cannot think of the problem right now. The question is ‘does Zeno ever catch the tortoise?’. I know the answer is supposed to be that he doesn’t but am convinced he would catch and overtake him.

not entirely sure what the link between Zeno’s tortoise paradox and this game theory puzzle is apart from they are both thought experiments?

For more information on Zeno and the tortoise have a read of this page:
mathacademy.com/pr/prime/art … /index.asp

  • ben

i agree, would he not overtake the tortoise?.. is this supposed to just be a paradox to make you think and i’m just taking it too literally :confused:

Mr. Black’s first shot should be aimed at Mr. White.
If Mr. Black’s first shot was aimed at Mr. Grey and he hits Mr. Grey, then there would be a 100% chance of being hit in the next shot, because Mr. White always hits his target and Mr. Black is the only target left.
If Mr. Black’s first shot was aimed at Mr. White and he hits Mr. White, then there is a 50% chance that he will get another shot.

It’s a distinction between language and science. If we give Achilles and the tortoise metre per second speeds, he’ll overtake. But if we identify that he’s “catching-up”, yet the tortoise is still moving, then it should be impossible for him to catch up because the gap, while ever decreasing, will never reach zero because there’ll always be a 1/infinity gap you have to overcome.

It’s kinda like the way the IMF and World Bank structure their reports. The South is growing economically, which is a great thing, but as a result of this the North gets to grow even more, widening the gap. But they someone manage to gloss over it with fancy graphs. Grr.