NOTE: This answer assumes we must deduce the restart times of the clocks without restarting them, and hence being unable to observe hands' behavior; I made this assumption so as to not trivialize parts of the puzzle.
How to fix the clocks:
The current time is
(deduced from the Pendulum clock)
To fix the Precision Clock,
wind the second hand backward until either the minute hand rests at the 12, or the hour hand rests at the 3 (since we don't know which hands are moving until we start); winding the second hand forward to the 12 from this intermediate position will result in the third (previously-unmoved) hand arriving at its correct place to show 3PM on the clock face.
Restart the precision clock at
To fix the Tilted clock,
turn the clock face so that the minute hand is pointing at the 12,
and restart the clock at
To fix the Relay Clock, simply restart the clock at
Total time to fix:
or "about an hour"
The second hand alternates direction every minute; the minute hand does so every hour; and the hour hand every 12 hours.
Given these, each hand gives
two possibilities for the current second, minute, and hour as follows:
either 26 (during even numbered minute) or 34 seconds (during odd numbered minute)
either 31 (during even numbered hour) or 29 minutes (during odd numbered hour), and since both possibilities are odd-numbered, the current time must be 34 seconds past a minute.
either 9 AM or 2 PM (but we just ate lunch, not breakfast, so the hour must be 2 PM; being an even numbered hour, this helps determine the full current time as being 2:31:34)
The minute hand moves 1 tick per minute, and the hour hand moves 1/12 tick per minute, so the gap between them grows at 11/12 ticks per minute (or 12/11 minutes per tick). The second hand moves at 60 ticks per minute, so the gap between the minute hand and the second hand grows by 59 ticks per minute (or 1/59 minute per tick).
Given the 18/18/24 tick intervals present on the clock between hands,
there must be a whole number gap of ticks between each hand at the time when the clock will be restarted.
Thus, starting from noon, there must be some (positive) number of minutes x that passes such that:
x can be written as y/59 for integer y, for the minute and second hands to have a whole number y of ticks between them
x can be written as 12z/11 for integer z, for the minute and hour hands to have a whole number z of ticks between them
Immediately, we can tell that
11 and 59 are relatively prime, so unless x is a whole number, y/59 and 12z/11 cannot be equal.
since 11x/12 = z, a whole number, and 11 and 12 are relatively prime, x is divisible by 12.
Thus, the time to restart the tilted clock is
some multiple of 12 minutes into an hour.
The true second hand must point at 12 for the displayed time to be a whole number of minutes into an hour. The true hour hand must be between 2 and 4 (10 and 20 ticks away) to match the working time period, and only an 18 tick gap (not a 24 tick gap) satisfies this need.
This position of the true hour hand determines the restart time:
3:36:00; we can confirm that, with the minute hand at the 36th tick, the hour hand 3 ticks past the 3, and the second hand at 12, this produces the 18/18/24 tick gap we see between hands of the tilted clock currently, and properly represents 3:36:00 PM, "about an hour" after we started working on these clocks.
The second hand being offset by +7 seconds while pointing at the 7th tick means we need to
set this clock to a whole-minute time (second hand at 12) and restart the clock.
This also means that the minute hand must
point directly at a tick in its correct position, and must be displaced a whole number of ticks from its current position.
Correcting the minute hand requires
the second hand to be wound a a whole number of ticks, and will then be a whole number of ticks away from the 12; winding in the opposite direction until the second hand reaches 12 will displace the hour hand a whole number of ticks to rest precisely on a tick. The minute hand must therefore rest on a minute that is a multiple of 12 in order to agree with the hour hand.
But, we are constrained in terms of which times we are able to produce on the clock's face according to the means by which it is to be wound. We can observe,
since "wiggling" the second hand back and forth by one tick will move the hour and minute hands at one tick each in opposite directions, that keeping the home position of the second hand fixed while "wiggling" produces minute/hour hand positions that are always symmetrical to each other across some fixed diameter of the clock.
In the hands' current position,
this "diameter" passes through the first tick past 8 (and through the first tick past 2 on the opposite end of the clock), but after winding the second hand back 7 seconds, one of the other two hands also moves seven ticks backward, which shifts the diameter of reflection back by 3.5 (half of 7) ticks; the true diameter of reflection then passes halfway between 7 and 8 (and halfway between 1 and 2).
We are constrained further that this time must occur within "about an hour" from now, so that we can restart the clock at the time we produce on its face within said period of time.
For the hour hand to point between 2 and 4 (roughly our working period), the minute hand must fall on a minute between 11 (55 minutes) and 1 (5 minutes) to satisfy the reflection constraint; the only minute value in this range that can be accompanied by the hour hand pointing directly at a tick is the 0 minute mark, or directly at 12. The reflection constraint then requires the hour hand to rest precisely on the 3, which properly produces 3:00, and falls within our working time period.
Starting from the initial hand positions, we correct the clock hands by
first winding the second hand backward until either the minute hand rests at the 12, or the hour hand rests at the 3 (since we don't know which hands are moving until we start); winding the second hand forward to the 12 from this intermediate position will result in the third, unmoved, hand arriving at its correct place to show 3PM on the clock face.
Immediately, this clock seems somewhat similar to the Tilted Clock in that:
1. We have no way to change the clock hands' positions, and
2. At any given time, one hand each points to the current second, minute, and hour
Given these, all we have to deduce is the proper time to restart this clock.
Beginning with the true hour hand,
which must appear after the 2:24 mark and fall within "about an hour" of 2:30, the apparent minute hand and the apparent second hand could each be the true hour hand, but the apparent hour hand cannot. Both potential true hour hands indicate approximately the same number of minutes past the hour (between 36 and 48), thus only the apparent hour hand can represent the true minute hand.
We'll estimate the two possible times on the relay clock as
2:41:18 and 3:41:13, each within "about an hour" of our current time.
A few further observations to assist us:
1. The clockwise order of the hands is fixed whenever two or more hands do not overlap; The apparent second hand will always be behind the apparent hour hand and ahead of the apparent minute hand, because any time a faster hand reaches a slower hand, their speeds switch and the order of hands is thus maintained.
2. During normal function of a clock, the clockwise ordering of hands is not fixed. At 1:01:00 PM, for example, the second hand points at 12, the hour hand points slightly ahead of 1, and the minute hand points 1 tick past 12. There are, therefore, some times of day that the Relay clock cannot fully represent because of the fixed hand ordering. In addition, even at times when the hands of a normal clock would appear in the same order as they do on the relay clock, the hands can be off by one hand position either clockwise or counterclockwise.
We can fully determine the displayed time on the clock by ascertaining
whether, at 2:41:18 under normal circumstances, this clock would show the current hand positions.
All hands start at 12 with their apparent roles.
No collisions occur in the first minute.
For each following minute until 1 PM,
the true second hand switches twice, first with the true hour hand (between 12 and 1) and then with the true minute hand (always past the true hour hand). Each such minute effectively "rotates" the hands between true roles (second hand => hour hand => minute hand => second hand). After three minutes (6 switches), all hands are at their original roles. This continues for 57 minutes until 12:58, again with all hands in their apparent roles.
In the next two minutes,
we have four switches occur as expected, with the last occuring precisely at 1PM (where the true second and minute hands meet at 12). After the first three switches, the apparent second hand is again the true second hand, and then switches with the apparent hour hand (actual minute hand) at 1PM, swapping them.
So to sum up, at 1PM:
the true second hand is the apparent hour hand pointing at 12
the true minute hand is the apparent second hand, also pointing at 12
the true hour hand is the apparent minute hand, pointing at 1
In the next minute,
the true second hand only switches with the true hour hand. For another four minutes, two collisions occur each minute, making nine collisions since 1PM (divisible by 3), so the hand assignments are the same as at 1PM.
During the sixth minute,
the normal two collisions occur as before between 1:05:05 and 1:05:06, then near 1:05:30, the true minute hand hits the true hour hand just past the 1 on the clock face.
the true second hand is the apparent minute hand, pointing at 12
the true minute hand is the apparent second hand, pointing a tick after 1
the true hour hand is the apparent hour hand, pointing at half a tick after 1
For the next 54 minutes, two collisions occur per minute, in true second hand => true hour hand => true minute hand order; 108 total collisions being a multiple of 3,
the hand assignments at 2PM are the same as at 1:06.
Next minute, only the true second and true hour switch. For 9 minutes, 2 switches each (18, divisible by 3)
so the assignments at 2:10 match those at 2:01, namely:
the true hour hand is the apparent minute hand, pointing almost a tick after the 2
the true minute hand is the apparent second hand, pointing at the 2
the true second hand is the apparent hour hand, pointing at 12
The true second hand switches with the true minute hand and then with the true hour hand within about a second of each other. Then near the end of the minute, the true minute hand switches with the true hour hand.
The positions as of 2:11 are:
the true hour hand is the apparent hour hand, pointing almost a tick after the 2
the true minute hand is the apparent second hand, pointing one tick after the 2
the true second hand is the apparent minute hand, pointing at 12
For the next thirty minutes, 60 switches occur in the order of true second => true hour => true minute => true second, so at 2:41:00, the positions match those at 2:11:00 above.
To reach 2:41:18, the true second hand switches with the true hour hand. This makes the true second hand the apparent hour hand, which does not match the current state of the Relay Clock.
This indicates that the time shown on the Relay Clock is in fact
so we will restart the Relay clock at that time.