It's a Puzzlement

Lights Out

The diagram below uses x marks to denote 10 lights out of 25 that are turned on in a five-by-five grid. You want to turn all the lights off. The switch for any light turns it on or off, and also changes the state of the (up

		X	X
X			X
X				X
	X	X		X
			X

to) four lights that are horizontally or vertically adjacent. For instance, if you operate the switch for the light in the very center of the diagram, it would turn that light on, it would also turn on the two lights to either side and the one above, and would turn off the one below it (which is now on). How do you turn the lights off? Optional: If all the lights were out, how would you turn a given one on?

Loaded Die

In the last puzzle, the problem was to select numbers at random from 1 to 6 using a die known to be "loaded." Chris Yaure suggested the following technique. Make a table giving a one-to-one correspondence between each of the six permutations of Medium-Low-High and the six digits. Throw the die, and note the number that comes up. Throw the die a second time. If this number is the same as the first, forget what has been thrown and start over. But, if the second toss gives a number that is different from the first, toss the die again. If this third toss is the same as either of the previous two, start over. If it is different from the previous two, use the ranking of the three numbers that have come up and the table to find a random digit.

Other solutions are possible, but I think Chris' solution minimizes the expected number of tosses per random digit. Another method is to toss the die six times. If all six tosses give a different number, then take the last number (or the first, or …). If two or more are the same, toss another six times. Note that it is not sufficient to toss until you have three (or six) in a row that are different.

Donald Behan noted that this problem is an Exam 3 problem and is similar to one found in Sheldon Ross's Introduction to Probability Models.

Bob Conger, John Herder, and Sebastien Millette also solved this problem.