Problem of the Week #13: Monday April 3rd, 2023 The tentative plan is to make this month's problems all about miscellaneous puzzles and games. I have some in mind, but if you have one you'd like to volunteer, please let me know. As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
A string of lights are each attached to a switch, but the switches are designed to change the state (turn on if it’s off, and turn off if it’s on) of the light it’s attached to as well as its immediate neighbours. For example, consider the string of five lights below (“X” denotes a light that is off, and “O” denotes a light that is on):
X – X – X – X – X
a – b – c – d – e
For this string of lights, pressing the switch labeled “c” would result in the lights looking like this:
X – O – O – O – X
a – b – c – d – e
And then pressing the switch labeled “a” would make them look like this:
O – X – O – O – X
a – b – c – d – e
a). Starting with all five lights off in the above string of lights, try turning on all five lights.
b). Find a starting state of the five lights (which ones are on and which ones are off) where regardless of what sequence of light switches we press we cannot turn on all five lights.
c). How many different starting states can there be for this setup?
d). How many of the starting states in part c) are ones where we cannot turn on all five lights?
Now consider connecting the two lights at the end to form a loop, like this:
e). If we start with this state where all five lights are off, try turning on all five lights.
f). How many starting states are there in this setup?
g). In how many of the starting states in part f) are we able to turn on all five lights?
h). Try questions a) to d) if instead of five lights we have n lights, where n is a natural number, in a string or a loop.
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.