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Comment by I_complete_me | Hacker News Reader

I think it was this book where I first saw the puzzle:

    Does New Year's Day fall more often on a Saturday or a Sunday?

Such a simple puzzle with to (then) me such depth of knowledge to uncover the answer.

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I don't see it there: https://archive.org/details/boris-a.-kordemsky-the-moscow-pu...

Do you recall the solution? It seems like a tricky problem. No approach occurs to me aside from a brute force analysis of the 400 year Gregorian calendar cycle, accounting for the complete leap year rules (under which, e.g., 1900 was not a leap year but 2000 was).

It turns out that New Year's Day is a bit more likely to fall on Sunday than Saturday. To confirm, here's a bit of simple JavaScript that checks a complete 400-year cycle. You can run it in your browser's console:

  const counts = Array(7).fill(0);
  for (let year = 2000; year < 2400; year++) {
    counts[new Date(year, 0, 1).getDay()]++;
  }
  console.log(`Number of Saturdays: ${counts[6]}; Number of Sundays: ${counts[0]}`);

Output:

  Number of Saturdays: 56; Number of Sundays: 58

Note: getDay() returns 0 for Sunday and 6 for Saturday (https://developer.mozilla.org/en-US/docs/Web/JavaScript/Refe...).

Unless you're considering a limited date range, it should fall equally often on both in the limit since neither 365 nor 366 (leap year) are multiples of 7.

So you’d think that, and it is approximately true. But the calendar repeats every 400 years (there are 97 leap years in 400, and 497 is a multiple of 7). And 400 isn’t a multiple of 7, so you can’t have it work out exactly.

But if the pair of (date, weekday) repeats every 400*7 years, then it still works out exactly?!

In the Gregorian calendar, January 1 falls on a Saturday in any year that is a multiple of 400. (This follows from any correct day-of-the-week algorithm, such as https://firstsundaydoomsday.blogspot.com/2009/12/quick-start....) Since each 400-year cycle begins on the same day, rather than balancing out, any imbalances would actually accumulate.

To check, see this 5-line brute-force calculation in JavaScript: https://news.ycombinator.com/item?id=42024474

Ah, the number of days in 400 years is 400 * 365 + 97 = (57*7+1) * (52*7+1) + (14*7-1) = 0 mod 7, so indeed it doesn't work out.

Exactly.

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