So last week was the first weekly Joburg Math meetup. Which basically means a bunch of friends meeting up and playing with math olympiad type problems. If anyone is interested these meetups happen on Sundays at noon in Motherland coffee Rosebank Mall, and of course everyone is welcome. hopefully they'll also give me new ideas for more regular math type blog posts.
Anyway here is a problem from The book Creative Mathematics by Alan Beardon, and soem generalizaions due to those involved in the meetup.
Problem 1. I have a line of n lightbulbs. Each lightbulb has a switch associated to it. The switch toggles (if on changes to off, if off changes to on) all bulbs except the one it's associated to. If all the bulbs start out as off, for whcih n can they be changed to all on?
Problem 2. What if I put the bulbs in an n1 by n2 grid. Again each bulb has a switch associated to it. The switch toggles all bulbs sharign either a row or coloumn with it's bulb. Another way to say that is that it toggles bulbs agreeing in with it's position in exactly one coordinate.
Problem 3. What if I have an n1 by n2 by... by nk grid. Now each switch toggles bulbs which agree in exactly k-1 co-ordinates.
Problem 4. We have the same grid as problem 3 but now the switches toggle bulbs agreeing with our bulb in exactly l coordinates for general l.
I won't say how far we got with these at the meetup because it would spoil the fun somewhat. I will say that only problem 1 actually occurs in Beardon's excellent book, so it's not to be taken as a given that there are nice solutions beyond that.