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!
For your chance at this week’s fabulous prize, please answer the following 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.
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%.
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.
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.)
As there are consecutive times that are valid reflections (i.e., 8:20 and 8:21), the “shortest” wait is 0 minutes.
Answer: 0h00m.
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.
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.
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 |
This post was last modified on December 16, 2017 6:54 pm
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