The blue-eyed islanders

Today’s Riddler problem is another classic. The current incarnation of the puzzle is about error-prone mathematicians, while the classic version is about blue-eyed islanders.

A university has 10 mathematicians, each one so proud that, if she learns that she made a mistake in a paper, no matter how long ago the mistake was made, she resigns the next Friday. To avoid resignations, when one of them detects a mistake in the work of another, she tells everyone else but doesn’t inform the mistake-maker. All of them have made mistakes, so each one thinks only she is perfect. One Wednesday, a super-mathematician, whom all respect and believe, comes to visit. She looks at all the papers and says: “Someone here has made a mistake.”

What happens then? Why?

Here is the solution:
[Show Solution]

The puzzle of the pirate booty

Today’s puzzle was posed on the Riddler blog, but it’s actually a classic among problem-solving enthusiasts, and is commonly known as the pirate game. Here is the formulation used in the Riddler:

Ten Perfectly Rational Pirate Logicians (PRPLs) find 10 (indivisible) gold pieces and wish to distribute the booty among themselves.

The pirates each have a unique rank, from the captain on down. The captain puts forth the first plan to divide up the gold, whereupon the pirates (including the captain) vote. If at least half the pirates vote for the plan, it is enacted, and the gold is distributed accordingly. If the plan gets fewer than half the votes, however, the captain is killed, the second-in-command is promoted, and the process starts over. (They’re mutinous, these PRPLs.)

Pirates always vote by the following rules, with the earliest rule taking precedence in a conflict:

  1. Self-preservation: A pirate values his life above all else.
  2. Greed: A pirate seeks as much gold as possible.
  3. Bloodthirst: Failing a threat to his life or bounty, a pirate always votes to kill.

Under this system, how do the PRPLs divide up their gold?

Extra credit: Solve the generalized problem where there are P pirates and G gold pieces.

Here is the solution to the main problem:
[Show Solution]

And here is a solution to the general case:
[Show Solution]

If this sort of problem interests you, I recommend taking a crack at the riddle of the blue-eyed islanders, or the unfaithful husbands. More information here as well.