bonanova

Nice. Curious about your algorithm. Mine ran forever with n=100, (and came up empty.) I tried direct solutions also, restricting the first to be square and the second to be 1xm. Came up empty again.

Wow. That is thought-provoking. This is one case for which I am glad the solution details are not (yet) divulged. Nice puzzle also.

So this is based off simulation? My method led to a different conclusion but it wouldn't be the first time I made a mistake. The numbers given were from a simulation. I tried calculating it exactly but with average rolls exceeding 4 the numbers got too big to count. The only surprise to me was the frequency of 6 being markedly higher than 5 or 7. BG needed a somewhat flat distribution preceding 12, and it turned out flatter than I anticipated. But his 6 5 4 3 2 1 argument would dominate small fluctuations.

Nice problem but just to clarify

I like this. A one dimensional jigsaw puzzle. I'll give it a look this weekend.

Step 1 would be the problem.

I won't spoiler this answer cuz it's offered in jest. Start instead with a line AB of length 1 and construct a line of length sqrt(1). Then scale up by factor of three.

More clue letters added. See above post.

I think you have to determine what the coins are to determine which way the head is facing. If the rotating coin were a US cent, then the head faces to the viewer's right. Nearly all other US coins (except for very recent mintings) have heads that face to the viewer's left. I suppose that you have to use the circumferences of the coins to find out which pair would satisfy the other conditions of the OP. I guess that's what was meant. I just pointed out that after a complete revolution the facing "way" is unchanged.

[spoiler='Slight change in the process does it']Have them stand in a straight line. Instead of requesting a show of hands, ask them to take a step forward. Liars will step forward at one or the other request. Truth-tellers from out of town will take two steps. Truth-telling locals wont' move. The one-steppers comprise all of the liars.

No need to spoiler. If it has made a complete revolution, (and it has) it's facing the same "way" after as before. Maybe I'm missing something.

There are a few functions that are their own inverses. 1/x (away from 0) is one instance. For tiebreaker are you asking for other instances of functions for which f[f(x)]= x? Or for more functions whose inverses are their reciprocals?

Good one. One to go. I'm amazed that #2 is the last one to be solved. I thought it would be first.

The OP does not say there is a third contest, but I can see how it's implied. "In a row" is redundant if not.

See OP. Identified answers already used.

“TBITL!” (7) ( D_ _ _ _ _A )