Minus' Solution to

Switch Which Way
Posted Weeks of Novmeber 2 - 15, 1997

If every switch is flipped twice, that means it is turned on and off. Turning each switch on and off will take care of six "switchings," and all switches will have been turned off as a result.
Here is one possible switching order:

switch A up
switch B up
switch C up
switch A down
switch B down
switch C down
 (Now you are back at the beginning,
  with all lights off and all switches down)
switch A up to turn on Light B
switch B up to turn on Light C

Paul Harrell, Jack Ulrich, and Jordan Thorndyke, from South Scotland School in Laurinburg, NC, tenderized this week's challenge with a big mallet and treated Minus to this succulent dish.

Egor first turns switch B up. This turns light C on.  Second, Egor turns switch C up and this turns light A on. Then he turns switch A up and this turns light B on.  Next he turns switch C down and this turns light B off.  After that, Egor turns switch B down which turns light A off.  Then Egor turns switch A down, this turns light C off.  Next Egor turns switch B up, this turns light C on.  Finally, Egor turns switch A up and this turns light B on!

This was a toughy, but we still had fun!

First I drew a diagram of which light comes on when a certain light is flipped up.  Then I drew another one showing which light comes on when each light is flipped down.  Then I started putting together letter combinations that did not have to lights that were  flipped down or up twice in a row. I also made sure that each light was flipped at least twice.

Following these rules I came up with the pattern, B A C A B C B A.  I then tried it out following my diagram and it was right.  It worked out according to the diagram and to the rules.  So, one of the answers to the MathDen Challenge is, B A C A B C B A.

1. Egor flipped switch A up which turns light B on.
2. He flipped switch B up which turns light C on.
3. He flipped switch  C up which turns light A on.
4. He flipped switch  A down which turns light C off.
5. He flipped switch  C down which turns light B off.
6. He flipped switch B down which turns light A off.
7. He flipped switch  A up which turns light B on.
8. He flipped switch B up which turns light C on.

Poor Egor has gotten himself into a mess. He decides to figure out, first, what switch turns which light on. He finds that  switch A, when up, turns light B on. When A is down C turns off. If Egor flips B up, C turns on. If B is flipped down, A turns off. When switch C is flipped up, A turns on, but when C is flipped down, B turns off.

All the lights are off now, and the switches are flipped down. Egor flips switch A up and light B turned on. With light B the only  one on, he flips switch B up to turn light C on. Egor now flips switch C up and turns light A on. Now, all lights are on and all switches are flipped up. It has taken 3 tries.

Next,  Egor switches A, B, and C down and turns all lights off. This has also taken 3 tries, which adds up to a total of  6 tries,  and poor Egor is right back where he started. Since he only wanted lights B and C on, he flipped switches A and B up. Finally!

Egor did it in 8 tries and only light B and C are on! The sequence is: A up, B up, C up, B down, C down, A down, A up, and B up.

To Minus, you never told us you had a girlfriend!!!!! But we love you all the same. Here is our solution:
A up = B on
A down = C off
B up = C on
B down = A off
C up = A on
C down = B off

Then we just rearranged this so that B and C were left on. This took eight flips of the switches. Of course we are right.

A up = B on
B up = C on
C up = A on
A down = C off
B down = A off
C down = B off
A up = B on
B up = C on

BEWARE THE POWERS OF VELCRO THE MAGIC COW!

They discovered that if Egor flicked switches A, B, and C up then lights B, C, and A would come on respectively. Then flick down switches A, B, and C and Lights C, A, and B go off respectively. Egor can then flick switches A and B up to turn on lights B and C.

T.L. Driggers mixed his answer up a bit--flick switches A and B up and lights B and C come on. Flick switch A down and light C goes off.  Flick switch C up and light A comes on.  Flick switches B and C down and lights A and B go off.  Flick switches A and B up and lights B and C go on.

If we were Egor, we would stay in another hotel next time!

Egor flips B up which turns C on.
Next he flips C up which turns A on.
Then he flips A up which turns B on.
Later he flips B down which turns A off.
Next he flips C down which turns B off.
Then he flips A down which turns C off.
Then he flips B up which turns C on.
Last he flips A up which turns B on.

Switch C Up = Light A On
Switch A Up = Light B On
Switch B Up = Light C On
Switch A Down = Light C Off
Switch B Down = Light A Off
Switch B Up = Light C On
Switch C Down = Light B Off
Switch A Up = Light B On

I found as many different combinations of ways of turning all the lights on. There was 6.  For the 6 ways of turning the lights on I found as many ways as possible of turning them off. There was less this time because there was a restriction of what letters could start. I then worked out the last section could either be ab or ba.  When I wrote a list I found 32 different ways.