Teaser 3023: Timely coincidence
From The Sunday Times, 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?
[teaser3023]
S2T2 
Jim Randell 7:35 pm on 28 August 2020 Permalink |
On any particular day (ignoring jumps in time, such as leap seconds, daylight savings time, calendar reforms, etc.) there is either a 1/86400 chance of observing the clocks displaying the same an any given second (assuming they clocks are equally likely to be observed at any particular second of the day – which is unlikely), or a 0/86400 chance if the date does not correspond to a valid time.
The following Python program runs in 100ms.
Run: [ @repl.it ]
import datetime from enigma import int2base, printf # consider a 400 year period, and count dates that give a valid time d = datetime.date(day=1, month=1, year=1900) i = datetime.timedelta(days=1) n = 0 t = 0 while d.year < 2300: if d.day < 24 and d.year % 100 < 60: n += 1 d += i t += 1 # output solution t *= 86400 x = t // n printf("p = {n} / {t} (~ 1 / {x}); date = {b}", b=int2base(x, group=2, sep=".", width=6))Solution: The daughter was born on 19th May 1961.
In order for the date to be read as a valid time we need the day of the month to be in the range 1-23, and the last 2 digits of the year to be in the range 0-59.
In each year of the required range there are 23×12 = 276 viable days.
In each 100 year period there are 60×276 = 16560 viable days.
In each 400 year period there are 4×16560 = 66240 viable days.
And in a 400 year period there is a total of 400×365 + 100 − 3 = 400×365.2425 = 146097 days.
Converting to seconds we get a chance of 1 in (146097 × 86400 / 66240) = 190561.304….
Truncating this result to an integer and reading as a date gives: Friday, 19th May 1961.
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