Lost solution

Mirroring the hands across a vertical axis of the clock face is equivalent to having a clock that runs backwards. So you have indeed found the time at which the hand spacing is exactly the same as in the starting position. But how would you go about proving that this is the first time at which the spacing APPEARS to be the same?

I still believe the puzzle as stated has no solution, or several solutions, depending on the assumptions made by the solver.

There isn t an earlier solution then this. Because all other rotations do not result in real clock times. And this one is the only one that generates a real clock time. Becuase the clock mechanisme is this the only one that the small arm is just at the same distance as in the starting position to the hour. 5 before three is equal to 5 past 9 as you look at how the arms stands. Only this mirroring works. All other times likes 20 sec to 4 or 20 seconds past 8 fails. The small arm must then be closer to the number then by the starting position. The other times around the half o Clock are even worse and generates no real clock times as you can see very easy. The more degrees you turn how worse they become. Only if you goes till this time.  Yes this clock time must be the and only solution. There is no other time so that the small arm is as close to three or the nine. The clock turn forward or backward exactly the same. So the three corresponds with the nine. You can t use hours closer to the twelve and also not further down to the 6 past the Three or nine at both sides. Because then you change the clock meganisme and the angels. I understands it and tried it out. Don t only know how to presice to explain it in English becuase it not my mother language but I try. I hope you understands me why other translations ( like in this picture ) not works.  You can take you watch and you will see it.

You know if you do not use for example the 9 or three our position you also do not have the 5 to or 5 past minutes position but for example 10 to or 10 past. Our quarter to or quarter past. And this infuence also the position from the our arm that then must be or closer or further away form the number. ( close to the hour ) this distence to the hour from the hour arm keeps with all the rotations and other translations the same but in the real time it will be never the same. It will Always differ. Except in the mirroring to the nine. This is the onl translation in wich this distence stays the same! I hope I say it clear now.

Back in post 34, sameez1 considered a clock whose face was divided into 60 units, where every unit represents one tick. The second hand ticks once a second, the minute hand ticks once a minute, and the hour hand ticks once every 12 minutes.

At 2:54:34, the second hand is at 34, the minute hand at 54, and the hour hand at 14. Exactly 120 degrees apart. This is Loyd's diagram:

At 3:38:58, the second hand will be at 58, the minute hand at 38, and the hour hand at 18. Exactly 120 degrees apart again. This is 44 minutes and 24 seconds away. This is the answer sameez1 recommended.

We might also consider 3:16:16, when all three hands point to 16. This is only 21 minutes and 42 seconds away.

I don't like this clock because I always thought Loyd would have gone for continuously moving hands without making us guess the clock mechanism. Also, it means that Loyd's diagram was not completely precise. (Loyd's second hand is not at 34, but is clearly between 34 and 35. Why did he not draw it exactly at 34? Was it so hard?)

But almost everyone else thinks these points are minor! If so, then I would endorse this clock and maybe Loyd's trick was all three hands lining up at 3:16:16?

This first time is not possible because it is not a clock time. When it is half past four. The small arm is between the 4 and then 5 just in the middle. After ten past this half time it is a bit further then the half more to te five then past the middle. Becuase it is an translation the hour arm keep still near the hour ( number ) and this is never the case as it is near half past the our. The time is no clock time. It means that the clock will never generate this time as you let it run.

As you can see on the picture that I show here with correspondent with your solution. It fails becuase the hour arm in this picture must be between the 4 and the 5 and a bit more to the five. Because its allreay past the half past 5.  On this way you can verifi and reason that all this rotations fails. Wich I did.  The minutes arm must Always be close to the twelve becuase the hour arm is close to the hour ( number ) so all the times around half past fails anyway.  But also 10 to or 10 past or 20 to or 20 past fails becuase the angel is diffent then with 3 0 clock. And that is why only this past 9 o Clock this mirroing wich I send is the only translating before after 12 hours later that is a real clock time that is possible. As you can verify. therfore I strongly believe And I am think I am right that only the horizontal mirroring that I gave is the only real solution. And there is no earlier solution that is in overeenstemming with how a clock runs. Translations are fun and the distances of the arms keep equal this way but also importent is that is give a real times that is possible when you let the clock run as was earlier mentioned by n95311 before. Thats why I took this in account too.

But what if I decide the hands appear equally spaced at 5:49:09? Then that time will be my solution to the puzzle as given. I can do that, because I'm the sole judge of how things appear to me.

Lol whats thats for kind of joke? Does the clock time if that is possible not count anymore?  Not serious anymore?

No joke. The puzzle asks for the next time the hands appear equal distances apart. The hands appear equal distances apart at 5:49:09, which is earlier than the time you have proposed.

Can you show me that time?

 Can you show me that time?

Set your clock at 3:15 let it run until 3:16:16 the hands all appear equally distant from each other.I don't know how you could honestly say that they don't.