Your mass will decrease by length^3, but the surface area of your hands will only decrease by length^2, so you will be able to climb up the sides just like an insect.

7. Suppose we have N companies, and we want to eventually merge them into one big company. How many ways are there to merge?

We must have 2 companies merged before we can merge a 3rd company, and and we must have 3 companies merged before we can merge a 4th, etc. For the first merger, there are C(n, 2) possibilities (n combinations of 2 companies). Then there are (n – 1) companies remaining since 2 have merged into 1. Of these (n – 1) companies, we can merge 2 at a time, C(n – 1, 2). That is the number of ways for the second merger, regardless of the number of ways for the first merger, so the possibilities get multiplied. There will be (n – 1) mergers, so

total # of possibilities = C(n, 2) * C(n – 1, 2) * C(n – 2, 2) * … * C(2, 2) = n! * (n – 1)! / 2^(n – 1)

Oh, I see that wikipedia already had the scoop at http://en.wikipedia.org/wiki/Manhole_cover

Quote:
“A round manhole cover cannot fall through its circular opening, whereas a square manhole cover may fall in if it were inserted diagonally in the hole (A _Reuleaux_ triangle or other _curve of constant width_ would also serve this purpose, but round covers are much easier to manufacture.) In reality, however, the existence of a “lip” holding up the lid means that the underlying hole is smaller than the cover, so that other shapes might suffice.”