247: Factoring the Time

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Factoring the Time
I occasionally do this with mile markers on the highway.
Title text: I occasionally do this with mile markers on the highway.


Cueball is bored and so he has been calculating the prime factors of the time shown on the clock. He's been doing this for almost two hours (from 1:00 pm to 2:53 pm). The number 2 is the smallest prime but is not a factor of 253, which is an odd number. The smallest prime factor of 253 is 11, which makes the other factor 23.

His co-worker decides to mess with Cueball and so he switches the clock from 12-hour time (2:53 pm) to 24-hour time (14:53). This makes factorization more difficult, as the time now shown is a four digit number rather than a three digit number. The number 1,453 is actually a prime number and so has no factors but one and itself; Cueball has only one minute to determine this. Note that the time to determine the prime factors of a number is the greatest when the number is itself prime.

Finding prime factors is not a trivial matter; the difficulty of factoring large numbers is the fundamental protection behind Public-key cryptography (but it uses much larger numbers).

This comic may also be play on the phrase "factoring the time", which means taking time to make sure one has sufficient time.

The title text refers to a similar challenge involving mile markers. At highway speeds (60+ mph), they would show up at least once per minute. Combined with the need to also concentrate on driving, factorizing numbers in the allowed time becomes much more difficult.


[One man is sitting at a computer. Cueball sits at a separate desk. There is a clock which reads 2:53.]
Cueball: 253 is 11x23
Man at computer: What?
Cueball: I'm factoring the time.
Cueball: I have nothing to do, so I'm trying to calculate the prime factors of the time each minute before it changes.
Cueball: It was easy when I started at 1:00, but with each hour the number gets bigger
Cueball: I wonder how long I can keep up.
[Man at desk reaches back and touches the clock.]
[Clock now reads 14:53.]
Cueball: Hey!
Man at computer: Think fast.


  • There is a widget for OS X to display the current time and its prime factors.

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I used to find the prime factors of the remaining distance until the next turn. It starts off difficult (for me) at 99 miles, etc. When it gets down to 30 miles, it gets easier. Then, at 9.9 miles, I have a tenth the time to factor 99 again, and it gets easier as the numbers get smaller. This is actually a pretty good way to pass the time while driving. (talk) (please sign your comments with ~~~~)

Paying attention to your driving might be a benefit. To yourself and others. Just sayin'.Jakee308 (talk) 20:00, 24 April 2015 (UTC)

I wonder how much time Randall spent trying to find a time that is not prime but the time + 1200 would be. -- Flewk (talk) (please sign your comments with ~~~~)

Took me about 5 minutes with a script after getting a list of primes from the internet: 1:19; 1:21; 2:09; 2:47; 2:53; 2:59; 3:23; 3:43; 4:07; 4:13; 4:27; 4:37; 5:33; 5:53; 5:59; 6:11; 6:23; 7:07; 7:13; 7:31; 7:49; 8:03; 8:17; 9:13; 9:31; 9:43; 10:03; 10:07; 10:37; 10:43; 11:11; 11:33; 11:39; 11:41; 11:47; 11:57 (also the technical cases of: 12:03; 12:05; 12:07; 12:11; 12:19; 12:41; 12:43; 12:47; 12:53) . -- 06:18, 20 January 2016 (UTC)

You know, in the state of Massachusetts, which is where Randall said he lives in the book What If?, mile markers on the highway are placed every 0.2 miles, so he would get only twelve seconds per marker if he's trying to do each and every one (less if he's slightly speeding like everyone else does when there's no traffic). 02:09, 22 February 2016 (UTC)

I added another explanation for the title text. It seems to me that factoring the (often same) mile marker numbers is a bit boring. Lanmi (talk) 11:55, 23 April 2016 (UTC) Lanmi

Why would factoring become secondary problem after switching to km?