A short small puzzle this week building off one interview concern I check out on the internet. The initial question to be this: |

At what times of the day execute the hour and also minute hand of one analog clock perfect align?

At first glance this could seem choose a trivial question, climate you realize that the hour hand moves smoothly and continuously roughly the dial (albeit at a slow pace than the minute hand), and does not snap to every quantized hour position on each hour. This complicates points a little, yet not as well much. (The answers space not 1:05, 2:10, 3:15 …)

It’s nice clear the the hand both align as soon as it’s exactly midnight (and midday). As soon as is the next time? It’s not 1:05, yet a tiny bit previous because, by the moment the minute hand is likewise at the 1 o’clock position, the hour hand will have progressed on slightly. The minute hand spins about the dial twelve time as quick as the hour hand (it completes one transformation in one hour whilst the hour hand move one hour, i m sorry is 1/12th that the clock face). |

In T hours, the minute hand completes T revolutions. In the very same amount of time, the hour hand completes the portion T/12 revolutions. Making use of degrees, we can see the the minute hand move at 360° per hour, and also the hour hand (360°/12) = 30° per hour.

Below is a graph reflecting the angle (in degrees) because that both hands for worths of T native 0 to 12. Whereby the lines intersect (example presented with the red circle), is wherein the hands will certainly coincide.

This happens at 11 various locations in between midnight and also (just before) midday, then repeated again an additional 11 times in the afternoon.

We can calculate the specific times by looking for the times as soon as the angle between the two hands is zero.

Let"s specify the angle HT, MT to be the angle (in degrees) the the hand (from 12 o"clock position), after time T (in hours).

HT = 30T

MT = 360T

For the hands to it is in aligned, the difference between the hour and also minute hand demands to it is in zero (after an arbitary variety of rotations), where n is an arbitrarily (integer) variety of rotations.

MT - HT = 360 × n

360T - 30T = 330T = 360n

11T=12n

We can discover the times, through insterting in n=0,1,2 …

T=12n/11

The hand overlap every (12/11) hours. Right here are the 22 outcomes (rounded come nearest second):

12:00:00 AM | 12:00:00 PM |

1:05:27 AM | 1:05:27 PM |

2:10:55 AM | 2:10:55 PM |

3:16:22 AM | 3:16:22 PM |

4:21:49 AM | 4:21:49 PM |

5:27:16 AM | 5:27:16 PM |

6:32:44 AM | 6:32:44 PM |

7:38:11 AM | 7:38:11 PM |

8:43:38 AM | 8:43:38 PM |

9:49:05 AM | 9:49:05 PM |

10:54:33 AM | 10:54:33 PM |

However, the concern I desire to ask is the next logical development of this problem. What if we include a seconds hand?

### Hour hand, Minute hand, 2nd hand …

Image: note Turnauckas | At what time of the day perform the hour, minute, and second hands every line up? |

As before, let"s define the angle HT, MT, ST to it is in the angle (in degrees) the the hand (from 12 o"clock position), after ~ time T (in hours).

HT = 30T

MT = 360T

ST = 360T × 60 = 21600T

For the hand to be aligned, the difference between pairs angles requirements to be zero (after an arbitary number of rotations).

MT - HT = 360 × n

ST - HT = 360 × m

(Where n and m are integer coefficients). Combine the equations us get:

360T - 30T = 330T = 360n

21600T - 30T = 21570T = 360m

These simplify:

11T=12n

719T=12m

Giving the result:

719n=11m

Both 11 and 719 room Prime, and also have no usual factors. For this reason (other 보다 the trivial instance of n=m=0), as n should be a multiple of 11, say n = 11x, for part integer x. Then m = 719x, and T = 12x. That reflects that the just time once this happens is after ~ an integer multiple of 12 hours, that is, at 12 o"clock.

All the locations where the hour and minute hand align (angle distinction being a lot of of 360°) are various from all the locations the hour and the 2nd hands align (other than 12:00).

What this means is over there that, various other than midnight (and midday), there space no various other times as soon as all three hands have specifically the very same angle.

(This answer makes sense once you think about it, because, if all 3 were come line-up, it has actually to occur when two line-up together well, so it would need to at one of the 11 time calculated previously with simply the hour and also minute hands, and also none of this correspond to location of wherein the second hand could be).

## How close have the right to we obtain them?A rapid search of internet reveals that Dr Rob, native The mathematics Forum additionally looked at this problem and found the the next he can get all 3 hands is at the time 5:27:27.3, as soon as all the hands room within a 1.0014 level sector. That goes on come remind us that this is more than likely still clearly shows to the nude eye together clock hands room thin, and the edge between 2nd marks is 6° |

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