Lost solution

@sameez1: It's almost 4:00 according to Dale's interpretation. The hours hand does not jump from 3 to 4 until the minutes hand has reached 12 o'clock. It is an analog clock but it works exactly like a modern digital clock e.g. the one on your computer screen, 3 hrs, 55 minutes and 35 seconds.

I have seen analog clocks that jump minutes or seconds, but never one that jumps hours which makes the explanation a bit strange. The math is quite believable though. May be there was an explanation about its operation in the original challenge.

@ Arisktotle this was presented in 1905 a genious ad campaign in my opinion.How many watches do you think they gave away?they offered 10,000.Where is Loyds answer?   Yes if you go with the course movement of  one complete rotation of the second hand to jump minute hand one minute (6 degrees) and 12 complete rotations of the second hand to jump hour hand 6 degrees (I have to apologize to JamesColeman if I use this)Then contrary to my first thoughts the time shown 2:54:34 is a perfect 120 degrees (20min)apart.And as Dale points out the next perfect time would be 4:00:40. I am glad to have this forum to discuss puzzles like these as most of the people I know would get bored with it and say who the---- cares. Thanks

No, it's 60 complete rotations of the second hand to "budge" the hour hand. When the hour hand does "budge", though, it goes directly from the '3' to the '4' in a single motion. So the hour hand does NOT move 5 times per hour; it moves once per hour. In the illustration, the hour hand has been "stuck" at the '3' for 55 minutes and 35 seconds (while the minute hand has been "stuck" at the '11' for 35 seconds).

I cannot think of a single clock or watch that actually does this.

On the inside of a watch, is a series of gears. All of the gears are connected to each other. As the second hand moves, it also moves the gear connected to the minute hand which in turn moves the gear connected to the hour hand.

Think about it. Have you ever actually seen an hour hand on a watch that jumps an entire hour immediately after passing 59 minutes and 59 seconds? No, you haven't. The hour hand is always moving constantly throughout an entire hour.

Mispost...oops

@ Cobra91  There is no clock that the hour hand jumps one hour all at once.  all you have to do is look at any clock the hour hand is very seldom pointing diretly at the hour.Are you honestly telling me this clock reads 3:54:34 not 2:54:34? What math were you using on your first post? I was using a too fine movement of half a revolution on the second hand which made it impossible to get perfect 20 minutes apart on all three hands.You look 12 minutes after on a larger clock where you can see the minute marks plainly and the hour hand will have moved a one minute increment off the hour. The next perfect 120 degree seperation is at 3:38:58

Yes, that's why I (and apparently everyone else besides NM Dale) was fooled by the puzzle. However, as noted by Arisktotle (post #23), digital clocks/watches DO operate precisely in this way, which is part of what makes them so much easier for most people to instantly read.

The clock reads 3:55:35. If you look at the diagram, you might notice that each hand is at most ~1 degree short of pointing directly at a particular number. The minute hand is FAR closer to pointing in the exact '55' direction than in the exact '54' direction, while the second hand is a LOT closer to indicating '35' than '34'. Also, the hour hand is not "6 degrees" short of the '3' as you claim; it is at most ~1 degree short of the '3'.

The slight offset is probably default for each hand, regardless of where they are.

Both of the mathematical proofs I referred to in that comment (post #3) were based on the assumption that all 3 hands rotated uniformly (exhibiting perfectly continuous motion). This is because I was deceived by Sam Loyd's superior cleverness and ingenuity, which allowed him to ensure his puzzle would be effectively unsolvable for simpleminded amateur mathematicians such as myself.

Please stop attempting to solve this puzzle and learn how to read a clock.

If the time were 3:55:35 then the hour hand would be pointing very VERY near to "4" on the dial.

Now, LOOK AT THAT CLOCK AND TELL ME WHAT TIME IT'S SHOWING

If this is a true analog clock, the angles between the hands may look the same but they are slightly different.  At 2:54:34, the minute hand is (34/60 + 4)/5 of the way between the 10 and 11,  or 91.33333...% of the way between the 10 and the 11.  The hour hand is (10.9133333.../12) of the way between the 2 and the 3, or 90.944444...% of the way between the 2 and the 3.  The second hand is 80% of the way between the 6 and 7.  So, all different angles.  Even if you slowly sweep the second hand from :34 to :35 and measure all the angles while you're doing it, they will never be the same because the angle between the minute hand and the hour hand will always be less than 120 degrees the entire time the second hand is between :34 and :35

Having meditated on it for a while, I concluded it is neither a standard analog clock nor a standard digital (behaving) clock. It is a mathematical clock. As in all (certainly old) watches, the minutes and hours hands are set manually and independent from each other. The puzzle is not concerned with what time it is at all. It just places the hands in suitable geometric positions and starts off the unwind. The challenge does not require anyone to know what time it is, just to find out what time elapses until the next perfect angular trisection. The only data we need  for that besides the image are the relative hand speeds and their relation to our time units: hours, minutes and seconds.

Which does not bring us a step closer to the solution, though I suspect it is Cobra's 12 hours!

By the way, this was not a hoax to keep the watches in the company's pocket. Ingersoll sold 50 million watches and was so dominant in the market that one US president was introduced on an event as the president from the country that produced the $1 Ingersoll watches! 10000 $1 watches was just peanuts to them! Chances are however that the marketing stunt failed or we would have known more about it and about the puzzle solution.

You're over-thinking it.

It's just a clock and certainly not a digital one.

@ Arisktotle you need to know the time it is to calculate the distance in time to the next position that the hands are again all an equal distance apart.the minute hand and the hour hand on a conventional clock are not independent of each other they are geared together at a 12 to 1 ratio.12 increments of the minute hand to 1 increment of the hour hand each increment is 6 degrees 1 minute.The second hand is free from that, it trips a cog that allows the minute hand to move one increment once in its revolution.the time is 2:54:34 the hour hand is pointing at 14min mark the second hand 34min mark the min hand at 54 min a perfect 120 degrees apart add 44minutes 24 seconds to that you have 3:38:58 thats the hour hand pointing at the 18min mark the min hand at 38min  the second hand 58min mark 120 degrees apart. At 3:00:00 the hour hand will be pointing at 15 minute mark at 3:12 the hour hand will be at 16 min mark at 3:24:the hour hand will be at 17 min  mark at 3:36 the hour hand will be at the 18 min mark won'tchange for 12 more minutes. So my answer to the puzzle has changed from those angle are impossible (Ithought the second hand was directly geared to the rest) to 44 min 24 seconds.

Clearly not, I was referring to an earlier post postulating that the clock could be behaving like a digital clock without looking like one, though I agreed it is a strange scenario.

Your assumption that the imaged clock is just a normal analog one is no more likely than any other explanation as the clock hands on the real time of 2 hrs 55 are definitely not 120 degrees apart. Sam Loyd was a well-known maker of mathematical puzzles (and chess problems) and would never permit such vagueness in his challenges.  

Every watch has a few knobs for initializing its use and for synchronization. These knobs set the hands independently from each other. I can set the hours hand on exactly 3 and the minutes hand on exactly 6, creating an impossible time. Obviously, after initialization, the hands no longer run independently but they will never display a correct time from that point onward!

The clock in the image does not reflect a real clock time either and I suggested the puzzle maker chose it for clarity (and for its proximity to a real time) in his mathematical puzzle - everyone can see the hands are precisely 120 degrees apart. Furthermore I argued that the actual time is completely irrelevant since the challenge is only concerned with the elapsed time after the clock moves on from its displayed state. The answer can be calculated by viewing it just as a surface with 3 rotating hands travelling at different speeds, ignoring the issue of whether or not the hands represent a "realistic time". Hence, a mathematical clock.

It is standard practice in the world of watch marketing to picture watches with impossible times! Almost all watches on the net aim to display 10 past 10 (psychology; associates with positivity and smiling) but many of them do so inaccurately. When you look closely, you'll see the hours hand is often too close to the 10 to match the minutes hand. So what Sam Loyd did to the puzzle watch is simply a trick of the trade.

@ Arisktotle You are taking me back to the thought the position is impossible to obtain.the hour hand and the minute hand are not independent of each they are geared together at a12 to 1 ratio. if you move the minute hand the hour hand moves with it if you move the minute hand and the hour hand don't move the clock is broke.The fact that you think that if you could move the hands independently you can create a position that is not a readable time proves that you agree.I was going with the word appears as Dark Army pointed out...So are you saying that there is no answer like I had originally thought? Lol my wife told me just go buy a watch its only a dollar.

I am not sure you can say the watch is broke. You can manually set an impossible time on any non-digital watch and then let it run; it is just part of the design.

And indeed, I agree that the time shown on this watch is impossible.

In spite of the fact that the time on the watch is impossible, it is possible to solve the problem. It is not necessary to read the time in order to solve it. You only need to set the hands in motion and see how much time passes until they make angles of 120 degrees again!

Similar situation: you can measure how much longer one man is than another without knowing how long each man is. That's why you also need not read out the real clock time in order to know how much time did pass.

I don't see how it could possibly be behaving like a digital clock. I doubt any clocks operated in such a way back in 1905.

Still convinced that you are over thinking this. The puzzle was made in 1905 and back then clocks were clocks. They didn't operate in unusual ways. If Lloyd knew that the clock did have a strange behavior, he would have mentioned it in the question as he has with other clock puzzles.

The hands are in the correct position to be 120 degrees apart. Aside from that, we shouldn't be taking the photograph literally. It's a poor diagram and probably a recreation.

Here is a photo of the original advertisement for the puzzle.

@ Arisktotle you need to mark the time to communicate how you arrived at how soon you thought all hands would again appear an equal distance apart. If you were communicating how you arrived at the difference in the mens height you could say i measured from the top of this mans head(a position) to the top of the other mans head (another position) The hands cannot be manipulated to read an impossible time without making them slip on the shaft they are mounted on example at 6:30 try to put the hands exactly together at the 6 can only be done if you force them to slip on the shaft they are mounted on.Then the watch would always be pointing to an impossible time. Did you look at the A question of time by Sam Loyd  that  James Coleman  suggested .Sam Loyd a genius beyond words for sure.   

I never thought it was operating like a digital clock (just like you), especially not the hours hand. It was the assumption I made in my first post to explain Dale's answer because all other answers were equally imprecise or unlikely. In my second post I made the assumption that it was sort of a mathematical clock (even better: geometrical clock) instead of a digital clock which makes more sense.

I don't agree you can pretend that the displayed hands are both (a) a really existing time and (b) all 120 degrees apart. This is mathematically impossible near the depicted hand positions and this is a mathematical puzzle. You can only solve it when you know precisely where the hands are supposed to be. if your calculation for the next occurrence starts from imprecise values then the seconds hand will easily be a few seconds off and that is a giant angular error! A picture by itself is never accurate enough and needs to be clarified by a correct "understanding" of what is meant.

Note that I wasn't "overthinking" the puzzle but "simplifying" it by showing that it doesn't matter whether or not the picture displayed is a real clock time. One can always calculate when the next 120 degrees angles will occur. In fact, you can place the hands anywhere on the dial at 120 degrees apart and the answer is still the same even when the time shown by the 3 hands is clearly nonsensical.

Btw, if you want to go complicated (which I won't) then you can assume that the clock 'ticks' which is the same as making one, two or all of the hands display some kind of digital behaviour. This will probably generate several different solutions to the puzzle.