What would you want to see at the Museum of Mathematics?

At the science museum in London they have this very cute little gadget used by mapmakers 150 years ago: an axle with a rubber ring around it, and the ring pressing against a cone. The whole lot is attached to a metal stylus; you trace around an area on a map with the stylus and a little reader tells you the area of what you've traced around. I always found that ingenious. The exhibit in London then goes on to show how you can use the same idea to integrate and hence solve differential equations, and finishes with a monster machine that can solve ordinary 4th order ODEs using basically the same trick; you set the coefficients with dials and then the machine draws a graph of the output. I'm afraid I know neither the name of the cute gadget nor the machine :-( but it strikes me as being appropriate for a "math museum"...

1) A high quality 3D movie (with glasses!) of sphere eversion, like this one

2) A Let's Make a Deal game show room, where people can play the game to death on a computer until they believe that they should switch doors. Offer candy prizes.

3) A scaled down Bridges of Koinsberg room, where you can try to walk across each bridge only once.

4) A large transparent (working!) replica of an Enigma machine.

5) A Velcro covered life-size Mobius strip which you can walk on with Velcro shoes (I hope you have good insurance)

A cool gadget I've seen in a few science museums: There is a vertical board with a lattice of nails in it. You drop balls in from the top, at the center. After dropping enough balls, you always see a Bell curve, "proving" the central limit theorem. Then a catch releases the balls, they are transported back to the top, and you start again. The cooler versions of this have the Gaussian predrawn in the background (which displays a certain level of confidence! And a willingness to replace missing balls).

Edit - This is sometimes called a Galton box.