k-man

But if you change the OP slightly to say that there are 7 truth tellers and 4 random speakers (instead of 6 and 5) then the solution works! (unless of course my program still has bugs ) I got 912 sets of answers with an identifiable truth teller and 112 sets of answers with no possible arrangements. Edit: grammar

You're right. I found a bug in my program (it's been a while since I wrote anything myself) where it was verifying the results. Now I see that there are 123 sets of answers that produce multiple arrangements with no way to identify a truth teller. Bushindo, is this the same answer you had in mind?

Edit: formatting

Curr3nt and Peekay got it!

You are standing in front of a L-shaped trench. The trench is 5m wide, very deep and continues as far as you can see in both directions. All you found lying around are 2 boards that are strong enough to support your weight, but unfortunately they are both just a bit shorter than 5m. You have no rope, screws, nails or any other way to securely connect the two boards together. All you have are these 2 boards. How do you cross to the other side of the trench?

Judging by that X-Ray... you're probably right

Curr3nt beat me to it

@Bushindo I agree with everything you said. The only thing I'm not sure about is why you are talking about distributions within limited intervals. The OP states that the function produces the numbers in the range (-inf,+inf). Of course, all the subintervals will fit that description, but I assumed that any real number can be picked even though the distribution is not given.

Your result is a bit different than curr3nt's, so I'd say at least one of you still has some bug(s) to catch

I think it's pretty clear that the formula is to calculate the maximum number of pieces one can get from n straight cuts. Sometimes people don't have enough time to provide full detailed description of their solution. I'm one of those people. That doesn't make their answer incorrect or not valuable. Given the general formula provided by witzar anybody can calculate the max number of pieces resulting from any number of straight cuts.

Fair enough. Here are my new picks given the new information. Can't say that I have experience playing draw poker with other people, so these picks are based purely on intuition without doing any calculations...

Usually in single-player poker (such as the one you can find on a slot machine) the payout is based on the TYPE of hand you have (Flush vs. Straight) without regard to the actual ranking of the cards involved in the hand (pair of Jacks pays the same as pair of Aces). Also, the amount of the payout grows non-linearly, so it is often worthwile taking higher risk for higher reward. For example, if I have a pair of Aces and K,Q,J of the same suit as one of the Aces it's certainly worthwhile dropping an Ace for a chance of getting a straight flush. You only have 1/47 chance of getting it, but if it offers 250/1 payout then it's well worth the risk. You also have a pretty good chance of getting a flush of a straight. Keeping the pair of Aces may give you higher probability of improving your hand to some other hand, but it prevents you from getting a straight flush. Since you ranked all winning hands linearly the results of the possible outcomes are not typical for the poker and therefore the experience and intuition are not of much help. Also, the picks made by your program as optimal for any given hand will not be optimal in a real game with non-linear payout schedule.

I wanted to post this solution, but the OP states clearly that you are picking up pieces at random, so feeling the edges is not allowed. It also only works for 3x4 puzzle, but fails for larger puzzles.

I don't really have the time right now to calculate, but if someone wants to try this, here is a suggestion to how this may be approached

There is a small flaw in your logic that doesn't provide for the maximum gain to the captain.

Thanks for a great puzzle and all the hints, Smith. Here is what I got...