GeekDad Puzzle of the Week Solution – Time for Reflection

This past week’s puzzle as previously posted:

As I sat in a conference room on Friday, I happened to catch the time of clock reflected in the tabletop surface. To my surprise, it made another valid time!

reflected331This got me to thinking about how often this occurs, the time difference between occurrences, and the time difference between the source and reflected time. And of course, you guessed it, I am mentioning this as it is this week’s GeekDad Puzzle of the Week!

For your chance at this week’s fabulous prize, please answer the following questions:

  • What fraction (or percentage) of the time does the reflected time reflect a valid time?
  • What fraction (or percentage) of these times are something different than the current time?
  • What is the longest wait between reflected times being valid?
  • What is the shortest wait between reflected times being valid?
  • What is the largest time difference between the actual time and its reflected time?
  • What is the shortest non-zero time difference between the actual time and its reflected time?
  • Does using a 24-hour instead of a 12-hour clock (ignoring AM/PM) make any difference to the above questions?

This puzzle was straightforward for most, once they figured out that the numbers did not change order, but simply went through the following translations:

0 -> 0
1 -> 1
2 -> 5
3 -> 3
4 -> X
5 -> 2
6 -> X
7 -> X
8 -> 8
9 -> X

The “X”s above show that certain numbers, namely 4, 6, 7, and 9 don’t reflect back into actual numbers. Therefore, any time containing a 4,6,7, or 9 can’t make a valid time when reflected.

For a time to be valid, on a standard clock. each digit has to meet the following criteria:

From 1:00 to 12:59, there are 720 valid input times. Of these, 480 contain at least one 4, 6, 7, or 9. Of the remaining 240, twenty-six have a 2 in the second digit, like 12:01. As this would translate to 15:01, it is not a valid time on a standard clock.

  • What fraction (or percentage) of the time does the reflected time reflect a valid time?

Answer: 210/720, or 29.17%This is different on a 24-hour clock. Through similar elimination (2 in the 3rd position is never valid, and only sometimes good in the second position), we get 330/1440 or 22.92%.

  • What fraction (or percentage) of these times are something different than the current time?

As 0, 1, 3, and 8 all reflect to themselves and 2, and 5 reflect to each other, only numbers with at least a 2 or a 5 in it would make a “different” time; times made up only of 0, 1, 3, or 8 would reflect back to themselves.Of the 210 valid times, 60 of them contain can be made from 0, 1, 3, and 8 exclusively.

Answer: 150 of 210, or 71.43%. Once again, this is different on a 24-hour clock. On such a clock, 96 valid times can be made from combinations of 0,1,3, and 8, so 234/330 or 70.91% of the times reflect a different valid time.

  • What is the longest wait between reflected times being valid?

Answer: 2h04m, from 5:58 to 8:02.On a 24-hour clock, there largest gap is 5h04m, from 18:58 (6:58PM) to 0:00 (midnight.)

  • What is the shortest wait between reflected times being valid?

As there are consecutive times that are valid reflections (i.e., 8:20 and 8:21), the “shortest” wait is 0 minutes.
Answer: 0h00m.

  • What is the largest time difference between the actual time and its reflected time?

The largest difference between the actual and reflected times on a standard clock is 3h33m. It happens at 2:22 / 5:55.

Answer: 3h33m.This is the same on a 24-hour clock, but happens a second time at 12:22 / 15:55.

  • What is the shortest non-zero time difference between the actual time and its reflected time?

Answer: 0h03m. This happens some 30 times throughout a day, whenever the first three digits are made exclusively of 0, 1, 3, and 8 and the minutes end in 2/5.

  • Does using a 24-hour instead of a 12-hour clock (ignoring AM/PM) make any difference to the above questions?

Answer: Yes, lots, as shown throughout above.Congratulations to Kurt Gaines, selected from among the reasonably well-reasoned answers to win this week’s fabulous prize. A $50 Gift Certificate from ThinkGeek will be on his way soon! Thanks to everyone that submitted an entry. Happy puzzling!

Position 1 Position 2 Position 3 Position 4
Blank 0-9 0-5 0-9
1 0-2

Judd is a life long IT and math geek, currently adding value in the area of digital analytics. Dad to both Max and Nora, who frequently star in the puzzles he writes for GeekDad, his dear wife Allison has learned to tolerate the constant puns.