**Which Switch? Brainteaser**

There are three on/off switches on the ground floor of a building. Only one operates a single lightbulb on the third floor. The other two switches are not connected to anything. Put the switches in any on/off order you like. Then go to the third floor to check the bulb.* Without leaving the third floor, can you figure out which switch is genuine? You get only one try.

**Editor's Note (10/20/14): This sentence was edited after posting for clarity. *

**Cryptarithm**

In these problems, each letter corresponds to a single digit. For instance, can you figure out which digit each letter represents to make the sum at the right work?

**Perplexing Polyhedron **

Can you picture a solid pyramid consisting of a square base and four equilateral triangles, alongside a solid tetrahedron with four faces identical to those of the pyramid's triangles? Now glue one triangle face of the pyramid to a triangle on the tetrahedron. How many faces does the resultant polyhedron have? It's not seven!

**Isogon Proof **

Think about the class of polygons known as serial isogons. All adjacent sides meet at 90 degrees, and the sides are of increasing length: 1, 2, 3, 4, and so on. The simplest isogon, with sides 1–8, is shown at the right. This is the only serial isogon known to tile the plane. But there are more isogons. Can you prove that the number of their sides must always be a multiple of 8?

**Unattacked Queens**

The properties of chess pieces play a part in many challenges, including in a group of problems about unattacked queens. Imagine three white queens and five black queens on a 5 × 5 chessboard. Can you arrange them so that no queen of one color can attack a queen of the other color? There is only one solution, excluding reflections and rotations.