It's a Puzzlement
Light Bulbs and More Unsolved Problems in Number Theory

by John P. Robertson

A room has three light switches, each of which controls one of three lights in another room. You want to determine which switch controls which light bulb. You cannot see (or otherwise detect) any effect of the lights from the room where the switches are located. You want to go once from the room with the switches to the room with the lights and determine which switch controls which light; you do not want to go back and forth between the rooms. These are normal up/down wall switches (no dimmers) and normal incandescent bulbs (not three-way bulbs for instance). You know that the up position turns the lights on and the down position turns the lights off. How do you determine which switch controls which light? Extra credit: how do you do it if you don't know whether up is on or off, but you know that it's the same for all three switches.

Unsolved Problems in Number Theory

The first problem posed by Gary Venter was to investigate sequences of consecutive integers such that each can be written as a product of two-digit integers. Edwin Jordan found several sequences of length six, starting with 779, 1022, 1271, and 3476. He and Brian Donlan found a sequence of length 8, beginning with 4895. Note that 4901 is 13*13*29, a product of three two-digit numbers. I think that this sequence of length eight might very well be the longest possible such sequence. There are theorems on the density of integers with only "small" prime factors, say primes less than 100, which show that the probability that larger and larger integers have only small prime factors declines very rapidly. Assuming statistical independence of such divisibility, one obtains a probabilistic argument against the existence of sequences longer than eight.

Due to complex mathmatical symbols, Mark Yasuda and Stephen Mildenhall's solutions to Gary's second problem are not available online. To receive the complete solution, which investigates ns such that (n+i)/i is a prime for i from 1 to , for k as large as possible, E-mail the CAS Office.