# Sunday Times Teaser 3023 – Timely Coincidence

### by Danny Roth

#### Published Sunday 30th August 2020 (link)

George and Martha possess two digital “clocks”, each having six digits. One displays the time on a 24-hour basis in the format hh mm ss, typically 15 18 45, and the other displays the date in the format dd mm yy, typically 18 07 14.

On one occasion, George walked into the room to find that the two “clocks” displayed identical readings. Martha commented that the long-term (400-year) average chance of that happening was 1 in just over a six-digit number. That six-digit number gives the birth date of one their daughters.

On what date was that daughter born?

1. Here is a manual solution.

Firstly, there are 97 leap years in 400 hundred years, which means that there are $$400 \times 365 + 97$$ days and hence $$146,097 \times 86,400$$ seconds.

On every day on which the date can be read as a valid time there will be a one second period in which the clocks match. So we have to count the number of times in 400 years that the date is valid as a time.

There will be 240 years in which the years will be valid as seconds. The months will always be valid as minutes. In each month there will be 23 days in which the day will be valid as an hour, that is, the overlap between 0..23 and 1..(29, 30, 31). So the total number of one second matches in the 400 years is $$240 \times 12 \times 23$$.

Hence the probability of a match is $$12 \times 23 \times 240 / (146097 \times 86400)$$ giving the inverse probability of a match as: $\frac{146097 \times 86400}{12 \times 23 \times 240}=190561.3043…$Hence the daughter’s date of birth was 15th May 1961.

2. Sledgehammer and nut?

You seem to know the number of seconds in a day.

“Leap seconds” probably do not matter.

We need to assume no changes to calendar before 14/9/ 2152.

3. I agree with Brian’s method. Over 400 year period there were some special years that were not leap years. Hours count from 00 to 23 but months start at 01, so they don’t always line up. Seconds count from 00-59, years from 00-99. So some years they don’t line up. Need probability, over 400 year period, of (Hr aligning with Yr)(Min aligning with Month)(Sec aligning with Day).

Working by hand, multiplying these 3 probabilities together, I get the same result as Brian.

4. Without needing to know the number of seconds in a day it is easy to produce the expression

in simplest terms (400×365+97)x30/23.

I am puzzled by the publication date October 04 2020.

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