So, this week I brightened up your life with this puzzle about light bulbs. My question is whether it is possible to keep flipping the switches such that all 4 light bulbs are switched on.
Shall we find out?
Well the answer to this puzzle is actually very simple: no. It is not possible to get all 4 light bulbs switched on (or off, for that matter) from this position. But why?
The reason has to do with a concept called parity – the simplest way this can be thought of is in terms of odd and even. Applying this to our problem, we can see that right now we have just 1 light bulb on, and this is an odd number. To solve the puzzle however, we would have to have 4 turned on, and this is an even number. Since our method of flipping switches always changes the states of 2 bulbs (an even amount) we can never change from the odd number we start with to the even number we need.
Confused? Look at it this way. We start with 1 bulb turned on. If we flip an adjacent switch, the currently lit bulb turns off while another bulb turns on, and we end up in exactly the same position. If we flip a non-adjacent switch, 2 bulbs turn on to give us a total of 3 lit, which is exactly the same problem as the original except all the states are inverted (1 off instead of 1 on). So we can never light all 4.
How’d you fare this week? Did you spot the solution, or did it evade you?