bonanova

Here's a refutation based on the impossibility of having chests whose value has uniform probability across the real numbers (or integers.)

I'll give a refutation by symmetry. Mainly because Bayesian formulas are opaque to me. (Read, I tried to understand a priori distributions once.) Your first choice is random. If you switch you end up the the choice you would have made, with 50% probability, without switching. Your expectation for either envelope is $1.5x, where $x is the lesser of the two amounts. OK, yeah, the faulty argument in my post above presumes a uniform probability on the real numbers for $x. Anyone who thinks $2 and $Graham's Number have equal probability needs to immediately make a random deposit into my checking account. (Account number supplied upon request.)

This was a fun puzzle.

Please do not post advertising links.

This is a classical paradox. I like this answer the best:

I assume "surrounded by" means "adjacent to." So do diagonals count?

Before reading the file I immediately thought of one method.

It depends on what is constant and what can vary. Do the edges all keep their original lengths? Does the cross section remain everywhere square? By "connecting 8" edges" do you mean deforming the shape and "gluing" the square faces to each other, a to b, b to c, c to d and d to a so that the new shape has no vertices (corners)? Do the 8" edges end up as circular arcs? What keeps us from doing this and then squashing the final shape flat so that its volume is zero? I can't visualize a process that gives a final shape having a definitive volume.

Are we looking at the back and front 4x4 faces from the same direction? Or, does left and right, as they refer to the back face, assume we have turned the shape around so we're looking at outside of the back 4x4 face? That is, if we label the vertices ABCD on the front face, clockwise starting from the upper left corner like this A B D C Then if the 8" edges connect these to the back face vertices EFGH as A-E, B-F, C-G, D-H would your connections be A to H, D to G, and so on? It sounds as if we are joining the front and back faces of a prism to form a torus that has a square cross section, only we're twisting the shape by 90 degrees before joining the faces. (Kind of like constructing a Mobius strip from a piece of paper. This new shape would have only three sides, just as a Mobius strip has only one side. Interesting.) I wonder, though, with the length (8) being only twice the side (4) whether this is even possible. One thought:

I noted the number(s) of the clue(s) that refer to each of the items. This is making it easier to piece together little clusters of relationships. Next I'm trying to fit these clusters into a table where one column is filled in (e.g. house numbers). Or probably into several tables, each with a different column filled in, depending on the "shape" of the clusters. It's a huge puzzle any way you look at it.

Thanks for the category lists and clarifications. This is actually a favorite style of puzzle, and I can see myself enjoying it for a while. @Thalia, Game on ...

You are given the following ten statements and are asked to determine a particular number. At least one of statements 9 and 10 is true. This either is the first true or the first false statement. There are three consecutive false statements. The difference between the numbers of the last true and the first true statement divides the number. The sum of the numbers of the true statements is the number. This is not the last true statement. The number of each true statement divides the number. The number is the percentage of true statements. The number of divisors of the number (apart from 1 and itself) is greater than the sum of the numbers of the true statements. No three consecutive statements are true. What is the number ?

@mmiguel, Nice.

@rocdocmac, Challenging puzzle! Clearly you spent some time on it. @Thalia, your questions (and the answers) were helpful. BTW, are you giving up on the puzzle? Have you listed all items by category yet? I'm in that process now, as a first step, before making a grid. Comments Assumptions: (just mention the ones that might be wrong) Questions:

Eight young people met at a party. After the revelry, each had fallen in love with one of the four persons of opposite gender and came to be loved by another of those four. Here are the particulars. John fell in love with a girl who is, unfortunately, in love with Jim. Arthur loves a girl who loves the man who loves Ellen. Mary is loved by the man who is loved by the girl who loved by Bruce. Gloria hates Bruce and is hated by the man who is loved by Hazel. So ... Who loves Arthur?

Both good answers. Here's another.

For convenience I'll post the puzzle here in text form.

Certainly. Let me know if you have any difficulty.

Can you find irrational numbers a and b such that ab is a rational number?

Agree. When I hit the send button, I realized my thinking was too simple. But instead of deleting my post (moderator privilege) I left it to take its licks.

I agree. I put it into his post.