bonanova

Right ... it's "discrete" differentiation. And that permits a (nice) direct calculation as opposed to simulation. I simulated the average first-duplicate position and got the kind-of well-known 23. I am modifying the program now to show the median value. The distribution is skewed, so the mean is slightly larger. And, I guess, the median must be the point of greatest slope of the cumulative probability.

Are we looking for a way that the four friends get a fair sized piece of the cake? Or, are we looking for a way to divide the cake exactly into fifths? To clarify what I mean, for two people to get a fair division of a piece of cake, the first one cuts it into two pieces; the second person chooses which piece to take. We don't get an exact 1/2 division necessarily, but it is a fair division. The OP seems clear that it's fairness we're looking for, so my question is probably not necessary. It's an interesting puzzle.

I'll mark this solved. Is there a more succinct solution?

But remember, I do not need a line. I just want the vertices that make the desired shapes. There is no request to actually connect the vertices.

With a careful reading of the OP can you do slightly better?

Consider the numbers from one to one million: 1, 2, 3, ..., 999998, 999999, 1000000. What is remarkable about the numbers 40, 8, and 2202?

Nice analysis! I wish I had thought of that.

Perhaps, sorry for any misunderstanding. No sarcasm was intended, nor would I want anyone to express sarcasm in this forum. Your question was proper, and my guess as to its answer was based on the fact that the total would be swamped by counting trivial permutations, while the count of combinations seems an interesting problem. I don't yet have the solution. SP has posted a clue. I'll leave the solution to you. Good Luck.

Here is a piece of plywood in the shape of an isosceles triangle. The side lengths are 1, 1, sqrt(2) units. Quick and dirty representation: A | \ | \ | \ | \ B--------------C The angle at B is a right angle. We'd like to cut this into two pieces of equal area. There are many ways to do this with a single cut. Which cut has the shortest distance?

You'll need a straight edge (ruler) also.

I would guess "ways" means combinations. The permutations of 271 1's are not that interesting.

In one village boys are desired, to work the land. And so couples are told to be sure they have a boy and then stop having children. In another village girls are desired, to increase the population. In that village, couples must bear a girl and then stop having children. The villages are of equal size, and heterosexual monogamy is practiced. By symmetry, there will a girl in each village for every boy in the other village. But marriages are permitted only within one's own village. What percentage of children in each village therefore can be expected not to find a mate?

Joe and Moe have nothing to do, so they take turns flipping a fair coin. Moe's mother interrupts their play by calling them inside for lunch. Given that Joe was the first to flip, what is the probability that Joe flipped more heads than Moe did when they stopped? When Moe's mother interrupted their play by calling them inside for lunch, one of them had flipped more heads than the other. Given that Joe was the first to flip, what is the probability that it was Joe who flipped more heads? w075

SP got there first. Can either of you convince the skeptics?

We can't make that assumption.

n! = 1 405 006 117 752 879 898 543 142 606 244 511 569 936 384 000 000 000 What is n?

You have a lighter and a single fuse that takes exactly 1 minute to burn. It is straightforward to time a 60-second or a 30-second time interval. How would you time a 10-second time interval?

I can mark only one post, so it's Rainman's, for getting the lowest sum. Rainman and SP finished off the rest of the analysis correctly. Nice work.