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 ...