This post has very little to do with reading, but it’s a really awesome trivia/math question to ask at your next party (or wherever you happen to be).

A good lead-in to this question would be after you and the opposite party are looking at an analog clock..

Here’s the question: **What is the first time after 9PM when the hour and minute hands of a clock are exactly on top of each other (that is, coincident)? **

First, consider working out this problem on your own… If you’re curious, hit the jump link below to find out about the solution…

#### Initial Impression

Your instant (perhaps even instinctual reaction) to the question above may be: 9:45PM! But would I pose such a question if the answer were so simple? Of course not. The key here is that by the time the minute hand gets to the 9, the hour hand of the clock will have moved a small amount (toward the 10). So one has to factor in a special “catch-up” time (perhaps a few minutes, to be calculated precisely below).

#### The Solution

Consider an analog clock that has markings of one minute between each hour label on the clock (i.e., there are five markings, in increments of one minute, between the 9 and the 10 on the clock). Now, as time passes from (9PM to 10PM), the minute hand moves as well, so the coincidence of the hour hand and the minute hand occurs sometime after 9:45PM. The key is that at all times, the proportion of the full sixty minutes traversed by the minute hand equals the proportion of the five increments of one minute traversed by the hour hand.

If we let **X** equal the minutes after 9PM to be the first time at which the hands are coincident, then we can set up a simple equation in which the proportion of the full sixty minutes traversed by the minute hand equal the “catch-up” time of the hour hand as a proportion of the five increments of one minute:

Namely,

** X/60 = (X-45)/5**

cross-multiplying, we get:

5X = 60X – 2700 –> 2700 = 55X,

and so, X = 49.090909 (or 49 + 1/11).

Putting the answer in the more elegant form of HH:MM:SS.mm, the answer is that the minute and the hour hand clocks are **coincident at 09:49:05.45** (approximated to the nearest millisecond). I got the answer in this form because X was how many minutes past 9PM the clock’s hands were coincident; the 1/11th factor we multiply by 60 minutes to get 60/11, or 5.454545…

#### An Easier Solution

There’s an easier way to think about this problem! First, note that the hour and minute hands are coincident exactly at noon and at midnight. In between noon and midnight, the time is cut into 11 equally-spaced blocks of time, each one 12/11 hours long. So that’s 12 hours * 60 minutes = 720 minutes / 11 intervals = 65.454545 minutes per each block, which corresponds to 1:05:27.27 per the HH:MM:SS.mm notation I used above (I ask the reader to verify that this answer is correct).

It follows that for the question: **What is the first time after 9PM when the hour and minute hands of a clock are exactly on top of each other? **all we have to do is multiply that answer by 9, and voila! I would do it this way: (60 minutes / 11 blocks) * 9 = 49.090909 minutes after 9PM (you can independently verify that it is equivalent to the approximate form of 09:49:05.45 PM).

#### Final Thoughts

I think this is a wonderful trivia question because most people won’t consider the movement of the hour hand whilst the minute hand is moving… Note that this question may be posed for *all times* other than midnight and noon, and the result would be found the same way. So if you’re partying it up late and it’s past 1AM (or 2AM), ask: “What is the *first* time after 1AM (2AM) when the hour and minute hands of a clock are coincident?” The answer: 1:05:27.27 and 2:10:54.55, respectively.

Of course, I make no guarantees how receptive the other party would be to answering this question. I thought it was a question worth considering, and I hope you found this post useful.

cool stuff to stimulate brain