As promised, an extension of the Island of the Logicians riddle. Intrepid reader John Rhoadhouse and I seem to have a solution (the credit is mostly his) but are not 100% convinced. So here you go.

As before, there are 100 logicians on an island. This time, instead of distinctively colored eyes, each logician has a dot, red or blue, on top of their head. Furthermore, the logicians are all different heights. Each logician can see the dot on the top of every shorter logician’s head, but not the dot on any taller logician.

As in the previous riddle, every night at midnight, a ferry comes and takes away any logician that knows the color of his or her own dot; and logicians don’t talk about the color of one another’s dots.

One day, in a crash of lightning, a stone tablet appears. This time it has two statements. Both are true. The first one is: “This information will enable all of you to leave the island.”

The riddle is, if you can, to find a second true statement that enables all of the logicians to leave the island, but not all on the first night. You may distribute the red and blue dots however you wish. If this cannot be done, and John and I believe it cannot, explain why.

If you’re having a tough time trying to wrap your head around this, you’re not alone. Consider the statement: “Exactly one of you has a red dot.” What you will discover is that if both statements on the tablet are to be true, only the shortest logician is allowed to have a red dot! Since the logicians know this, they are all able to leave the first night.

It’s maddeningly tempting to forget about the first statement on the tablet, especially since it’s easy to come up with configuration/statement pairs that allow everyone to leave the island.

This seems to be related to the paradox of a professor who promised to give his students a surprise quiz at some point the following week. The students’ reasoning goes like this:

1. The quiz can’t be on Friday, because if Thursday rolled around and the quiz still hadn’t been given yet, it wouldn’t be a surprise.
2. So the quiz can’t be on Thursday; because if Wednesday rolled around and the quiz still hadn’t been given, since Friday has been eliminated, the test would have to be on Thursday. But then the quiz wouldn’t be a surprise.
3. So the quiz can’t be on Wednesday, because if Tuesday rolled around and the quiz still hadn’t been given, it would have to be Wednesday because Thursday and Friday are out. But then the quiz wouldn’t be a surprise.
4. …And so on for Tuesday and Monday, and the students determine that a surprise quiz cannot be given at all.

This paradox derives from an unusually literal meaning of the idea of “surprise,” but has the same idea that the advance knowledge that something will happen affects its ability to happen in that way.

Another riddle for today. If you’re more interested in the board games than the riddles, my apologies. I haven’t been able to do much board gaming lately due to various irritating life factors. Following GenCon I hope to change that around.

In many ways this is the opposite of the Rope Riddle yesterday. The rope riddle has a very elusive solution, but the solution is crystal clear once you have it. By contrast, the solution to this one is pretty easy to get, but almost paradoxical in its nature. You may have seen a version of this riddle before. It’s the lead-up to a new riddle Thursday, invented by intrepid reader and riddler John Rhoadhouse. (Thursday’s is fierce and even more paradoxical than this one!)
On a certain island live 100 logicians. Some of the logicians have brown eyes and some blue. There are no mirrors or other reflective surfaces, and the logicians never discuss one another’s eye color. Thus, every logician knows the color of everyone’s eyes but his own.

Every night at midnight, a ferry comes to the island. Any logician who knows his eye color gets onto this ferry, leaving forever for the logicians’ paradise. The other logicians will wake up to find them gone. However, because of the lack of mirrors, this has never happened as far as the logicians know.
One day, in a crash of lightning, a stone tablet from The Powers Of Truth lands on the island. Its inscription: “At least one of you has blue eyes.”

Question 1: Who leaves the island, and on what day?

Question 2: The tablet doesn’t say anything that the logicians didn’t already know. How, then, does it help them to leave the island?

P.S. Without loss of generality, in the case where none of the logicians have blue eyes, the tablet says “At least one of you has brown eyes.”

GenCon Indy is next weekend, and I’m going. Due to work considerations I’m driving down either Friday after work or Saturday.

This one kind of sneaks up on you. I’d read it three or four times and never had an answer. Then reader John Rhoadhouse posed it to me the other day, and somehow I got a solution in only a couple of minutes; in another minute, he had a different, more elegant solution that I suspect is actually the intended one.

So here you go. Consider a certain kind of rope. It has the following characteristics: If you light one end, it will burn up in exactly one hour. But the rate of burning may vary unpredictably; for instance, the first half of the rope may burn in 10 minutes and the second half take 50; or it may be divided into many different areas; and there’s no way to tell. The upshot of this is that you can’t measure and cut the rope to measure a fraction of an hour.

Given two of these ropes, how can you measure 45 minutes?

1 (Not too hard). On every day of a given season, player A has a better batting average than player B. Does it follow that player A has a better batting average than player B over the whole season? Prove or give a counterexample.

2. (I don’t know the answer to this one.) At some point during the season, a player has a batting average of less than 80%. At some later point, that player has a batting average of greater than 80%. At some point, his average must have been exactly 80%. (If you don’t believe me, try it.) Why is this the case? What other percentages is this true for?
P.S. I don’t know why his batting average is so high. Maybe this is some weird hitter-friendly version of baseball.

Another riddle for today. I may have already shared this with some of my readers. This is one of my favorites and was supposed to go up earlier today, but something with my computer or WordPress ended up flaking out…

Five pirates have to divide a treasure of 1000 coins. The pirates have a ranking from highest to lowest. This is the system the pirates use to see who will get how much treasure:

1. The lowest ranked pirate proposes a way to divide up the treasure.
2. The pirates vote on the distribution scheme.
3. If more than half of the pirates vote yes, the treasure is divided as planned.
4. Otherwise, the pirate who proposed the scheme is forced to walk the plank, and the next lowest ranked pirate proposes a scheme.

The pirates have the following traits:

• They are perfectly logical.
• They are greedy and will vote to maximize their own treasure regardless of all other considerations.
• They are bloodthirsty and, all other things being equal, will vote against a proposal.

What should the lowest ranked pirate suggest?

Hint 1: Usually I would say something like “Now generalize the solution to any number of pirates.” But for this riddle, the solution is trivially generalizable once you have it.

Hint 2: The solution makes sense, but the distribution that it gives is pretty startling.

Other Notes: Someone asked me how the pirates came up with such a system. I wasn’t there, of course, but here are some speculations:

• There was grog involved?
• Or maybe an ancient curse?
• This is the way all pirates distribute treasure; it works better for non-perfectly-logical pirates?

A three part riddle today.

Part 1: You have 100 teammates, all standing in a row. They are facing forward so each can see the ones in front of him. Each one has either a red hat or a blue hat. Starting with the one in back and proceeding forward, each is asked the color of the hat on his head. If the guesser is right, the team gains a point.

The team gets a chance to decide on a strategy beforehand. However, the gamemaster is aware of their strategy, so they need to optimize the worst-case scenario. What strategy should they choose?

Part 2: Just like part 1, but work out the solution for any number of colors of hats.

Part 3: Like part 1 (only two colors of hats), but one of the teammates is a traitor. He is in league with the gamemaster and will do anything possible to mess up the team’s score. To make matters worse, the gamemaster, being aware of the team’s strategy, can place the traitor where he will do the most damage.

The players are aware of the traitor, though. What strategy can they use to optimize the worst case?

Today, a riddle.

I issue you and a partner the following challenge: I hand you five cards from an ordinary deck (no jokers). You stamp the numbers “1” “2” “3” and “4” on four of the cards, and hide the fifth. The choice of which to write on and which to hide is yours. You then leave the room and your partner enters. Given the four marked cards, your partner must guess the identity of the hidden card.

What strategy do you and your partner agree on beforehand that allows you to determine the hidden card? There is no need for trickery like imparting some meaning to the location of the mark on the card.

Warning: As it turns out, there is a lot of wiggle room in the solution. In fact, a solution exists even if the deck has 120 cards! Despite this, the riddle is strikingly hard. To my shame, I had to look up the answer, because after several hours, it was keeping me from getting any work done at the office. A mathematical background is not needed for the 52-card riddle, but it’s probably very useful if you try for 120.

Hint 0: A solution exists.

Hint 1: If you are like me, you will be tempted to throw away the ideas of suit and rank and just index the cards 0-51. While there is technically nothing wrong with this, starting with the cards divided into 4 groups is actually very natural and lends itself to an elegant solution.

Thanks to reader John Rhoadhouse for his help and thoughts in this and other riddles.

In other news: By popular demand, the return of 1KBWC next Sunday! If you are interested, let me know.