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My friend gave a question to me. I am not sure if it is an original puzzle, but he has 2 lines of Christmas lights and 3 circular timers with 96 switches apiece. Each switch turns on power for 15 minutes.

 

How can you arrange the timers, switches and lights — as far in advance as possible — so that lights would be on for 24 hours on Christmas day?

  • All lights are off for the last 15 minutes before Christmas.

  • For the next 24 hours, some lights are on at each point in time. (Some may also be off.)

  • All lights are off again for the 15 minutes after that.

No constraints outside of the 24 1/2 hours just described, except:

  • Any 24-hour interval before Christmas has all lights off for at least 15 minutes.

One timer may plug into another.

The timers operate as in this video. I am sure a lot of you have seen them in real life.

Note: This gem of a puzzle was given a make-over after a year and a half of being closed. The original poster might not be around to officially accept a solution. (Attempts were made to reach the poster half a year ago.)

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  • $\begingroup$ The timers only move when the power source is on, correct? $\endgroup$
    – user88
    Apr 7, 2015 at 21:17
  • $\begingroup$ Actually I don't know. I remember setting one up recently and it wasn't ticking while I was adjusting it plugged out. So I'd say you are correct. $\endgroup$
    – dzh
    Apr 7, 2015 at 21:32
  • $\begingroup$ How do these timers work? Are they a voltage source when on and a short when off? $\endgroup$
    – xnor
    Apr 7, 2015 at 23:17
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    $\begingroup$ I think it isn't possible. in order to power something for 24 hours a timer should have all switches on. In order for the timer to work the timer that's behind that timer needs to be on also for 24 hours and so on. So if you want at least 24 hours you can only make it work by have the power on all the time $\endgroup$
    – Ivo
    Apr 8, 2015 at 7:21
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    $\begingroup$ If someone solves this for "lights are only on during the month of December", can you let my neighbours know so I don't still see their lights on in April when they're still down south? :p $\endgroup$
    – Ian MacDonald
    Sep 28, 2016 at 12:46

1 Answer 1

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Here is a way to set up the timers 95 days and 23 3 ⁄4 hours before Christmas.

Timer A powers one line of lights and is off only for 3 intervals: 1, 3, 96.
Timer B powers timer C and is on only for 5 intervals: 1, 2, 3, 4, 96.
Timer C powers the other line of lights, advances only 5 intervals per day, and is on only during 3 intervals in its slow-moving schedule: 1, 3, 5. 

               Timer A  | - + - + + + ... + + - |   ---->   Lights A
                        | 1   3              96 |

               Timer B  | + + + + - - ... - - + |   . . .
                        | 1 2 3 4            96 |        '
        . . . . . . . . . . . . . . . . . . . . . . . . .'
       '
       '. . .  Timer C  | + - + - + - - ... - - |   ---->   Lights BC
                        | 1   3   5             |
Begin all timers at interval 2. They will combine to turn lights on for 24 straight hours when all three timers’ intervals align at 1, almost 96 days later.

This sets up a stencil of holes in timer A that is only filled completely for one day while timers B and C complete a full cycle of combinations, which takes 96 days to repeat.

Trying for that 96-day cycle, by having timer B advance of timer C daily by a small number of intervals that is prime relative to 96, the following template takes form.

             Day -1 |       Christmas       | Day +1

Overall lights   off| --------- on -------- |off


  Lights A       OFF|off? ? ? + + ... + +off|OFF      A's 96 & 1 must be OFF for
   Timer A        - | - ? ? ? + + ... + + - | -       day -1 to end correctly and
                ____|_______________________|___      day +1 to begin correctly
  1/4 hour       96 | 1 2 3 4 5 6 ...    96 | 1


  Lights BC      off|ON ? ? ?     ...     ON|off      C's 5 intervals during
   Timer C        - | + ? ? ?     ...     + | -       B's complete cycle of 96
Interval C       96 | 1 2 3 4     ...     5 | 6       is a coprime relationship
                    |                       |
   Timer B        + | + + + + - - ... - - + | +       B's 1 & 96 must be ON
                ____|_______________________|___      because A's 1 & 96 are off
  1/4 hour       96 | 1 2 3 4 5 6 ...    96 | 1

Fortunately the 3 ? ? ? intervals in timers A and C have only a few possibilities and one of them works out. This plays out as follows, skipping stretches of repetitive days.

Lights A and timers A and B do not vary from day to day.

         Lights A  | + +   + + + ... + +   |   +   + + + ... + +   |
          Timer A  | - + - + + + ... + + - | - + - + + + ... + + - |

          Timer B  | + + + + - - ... - - + | + + + + - - ... - - + |
                   |_______________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |


Days -96,-95
          Overall    | --- 23 3/4 h ON --- |off off     on      off| ("off off"
         Lights A    | +   + + + ... + +   |   +   + + + ... + +   |  is really
         Lights BC   |   +               + |                       | "off on off"
          Timer C    | - + -             + | - - - -             - |  here and
       Interval C    | 2 3 4             5 | 6 7 8 9            10 |  elsewhere)
                    _|_____________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |

Days -94 through -78 are essentially identical to day -95.

Days -77,-76
          Overall  |off off     on      off| on off     on      off|
         Lights A  |   +   + + + ... + +   |   +   + + + ... + +   |
         Lights BC |   +   +               | +                     |
          Timer C  | - + - +             - | + - - -             - |
       Interval C  |96 1 2 3             4 | 5 6 7 8             9 |
                   |_______________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |

Days -75 through -59 are essentially identical to day -95.

Days -58,-57
          Overall  |off on      on         |off off     on      off|
         Lights A  |   +   + + + ... + +   |   +   + + + ... + +   |
         Lights BC |     +               + |   +                   |
          Timer C  | - - + -             + | - + - -             - |
       Interval C  | 95  1 2             3 | 4 5 6 7             8 |
                   |_______________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |

Days -56 through -40 are essentially identical to day -95.

Days -39,-38
          Overall  |off off     on      off| on on      on      off|
         Lights A  |   +   + + + ... + +   |   +   + + + ... + +   |
         Lights BC |       +               | +   +                 |
          Timer C  | - - - +             - | + - + -             - |
       Interval C  | 94 96 1             2 | 3 4 5 6             7 |
                   |_______________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |

Days -37 through -21 are essentially identical to day -95.

Days -20,-19
          Overall  |off off     on      on |off off     on      off|
         Lights A  |   +   + + + ... + +   |   +   + + + ... + +   |
         Lights BC |                     + |   +   +               |
          Timer C  | - - - -             + | - + - +             - |
       Interval C  | 93   96             1 | 2 3 4 5             6 |
                   |_______________________|_______________________|
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 |

Days -18 through -2 are essentially identical to days -95,-1,+1.

...until...

                               Day -1              CHRISTMAS         Day +1

          Overall  |off off             OFF| ---- 24 hours ON ---- |OFF off
         Lights A  |   +   + + + ... + +   |   +   + + + ... + +   |   +   + ...
         Lights BC |                       | +   +               + |
          Timer C  | - - - -             - | + - + -             + | - - - - ...
       Interval C  | 92   95            96 | 1 2 3 4             5 | 6 7 8 9
                   |_______________________|_______________________|____________
         1/4 hour  | 1 2 3 4 5 6 ...    96 | 1 2 3 4 5 6 ...    96 | 1 2 3 4 ...
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