bushindo

I think there is a discrepancy with bonanova's answer because...

Excellent work so far. Sorry for not responding earlier. I was on a vacation and didn't have access to a computer

You are approaching this from the correct angle

I can't find a neat analytic solution for the problem of finding the expected number of steps until the ant leaves the circle, though it could be done by simulation or numerical approximation. This puzzle, however, asks a different question...

That is correct. I'll edit the OP to clarify that point. Thanks.

This is an extension of Prime's excellent puzzle What is the smallest 101-plus-digit number consisting of only 4's, 5's, and 7's that is divisible by '7777.....777' (one hundred 7's)? The solution *must contain* at least one 4, one 5, and one 7. EDIT: clarified the properties of the required solution

The two possible arrangements are

Great job, k-man. You got it.

Comments Doh! I went through everything again and found a couple of mistakes, so now the odds and expected payouts are as follows... Comments

Answer

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Excellent analysis, Prime. I just have 1 minor thing to add

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Sorry about that, my bad. Since D has the lowest probability in the above table,

Some comments

Definitely, k-man. Clarification are as follows

Almost there... Ah, more twist and turns. Thanks for such a great puzzle.

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Answer If the two logicians are required to give up if they figure they would never know it, then the game would go as follows I see what you're saying

Answer If the two logicians are required to give up if they figure they would never know it, then the game would go as follows

Answer

Approach

Here is yet another puzzle based on Suppose that there is a game as follows * There is a host with 10 stamps, 5 red and 5 blue. There are 4 players- A, B, C, and D. * In the beginning, the host affixes two stamps to each of the 4 players' head. The choice of stamps for each player is completely random (i.e. the host puts all stamps into an opaque bag and then draws them one by one). The remaining 2 stamps go into the host's pocket. Each player can see the stamps on the remaining 3 players, but can not see his own stamps nor the two in the host's pocket. * Starting from A to D (and then looping back to A and so on), the host asks if each player definitively knows his color (RR, BB, or RB). If the player does not know, the host goes on to the next player. First player to know his color wins. No guessing is allowed. Suppose that the host likes you, so he secretly offers you a side bet before the game. You have to pay the host 1 dollar before the game starts, and then you can choose whether to be A, B, C, or D. The payout by position if you win is as follows A: 3.5 dollars B: 2.5 dollars C: 6 dollars D: 7 dollars Which position should you choose for the greatest expected winnings?

See the part in red. My calculations show that the players would definitely end the game during that phase for the starting case of all RB stamps. I'm confused, as often. I thought we were facing one particular run of this game, and we are seeking the one case and reveal for which A does NOT announce his stamps at step 6. The phrase "starting case of all RB stamps" makes me think we have to make a general solution for all runs of the game. I see your point. The confusion is entirely my fault. I revised the OP to remove this confusion.

See the part in red. My calculations show that the players would definitely end the game during that phase for the starting case of all RB stamps.