You have one (1) ordinary six-sided die (i.e., the traditionally marked, uniform-density, cubical kind that you could find in any Vegas casino within about 5 minutes).

You need to make a random choice between four (4) alternatives, same probability for each (25%, i.e., barring the occasional coin-landing-on-edge weirdness that supposedly can happen in real life, which we won't worry about).

You could, e.g., roll the die and if you don't get 1,2,3,or 4, just roll again and keep rolling until you do. That would be too easy, and also could conceivably take a while if you get unlucky with the 5s and 6s.

But the

**real problem** is that, due to various contrivances entirely beyond your control, it just so happens that if you roll the die a second time, you will be plagued by frogs, and (trust me on this)

you really do **not** want to be plagued by frogs.

So you absolutely have to do it in one (1) roll. Good luck.

Well okay, if that one was too hard, here's an easier one:

You're running a meeting and there are four people who want to speak. You need to randomly decide what order they go in and you want to be Completely and Utterly Fair about it ... meaning each of the (4!=24) possible permutations of the speakers has to be equally likely. Once again, you have one (1) six-sided die, and the frogs will only allow it to be rolled once. Good luck.

Hint that probably completely gives it away if the last paragraph didn't:

**Where** are you rolling the die?

And a solution:

Find a stiff, flat piece of cardboard and give it a 90^{o} fold. Lay it down with the crease parallel to the ground and support it so that the flaps point into the air at a 45^{o} angle.

Or you can use a suitably propped-up gamebox lid.

Roll the die so that it lands in the crease.

For each of the 12 edges that can land in the crease, a different pair of numbers will be facing up. If one of the numbers is 2 and other is not 6, pretend the 2 is a 0. Subtract (smaller from larger) to get a number between 1 and 4.

For the second problem, it helps to imagine each of the four pairs of opposite corners of the die to be painted with a different color, say, 123&654 are red, 124&653 are blue, 153&624 are green, 154&623 are yellow.

Or just go ahead and paint it already; the frogs won't mind. Associate a speaker with each color.

Set up the same crease as before but this time, label one end of the crease as being the "north" end. Now we not only care which edge we land on, but which of the two possible ways the cube is facing. So that's 24 possibilities in all.

We look at the north face. Color of the top vertex goes first; go clockwise around the edge of the face to get the rest of the ordering. Done. The frogs are happy.

wrog