A most ingenious paradox

Here is a puzzle that I posed to Jonah last night, and then helped him to solve in fulfillment of his required ten minutes of nightly math homework:

What is the date (day, month, and year) at the beginning of The Pirates of Penzance?

Solution follows.

The Pirates of Penzance is the story of Frederic, a reluctant pirate’s apprentice. As soon as he turns 21 his apprenticeship is over and he promptly leaves his pirate comrades for a more virtuous life on land, where he meets and falls in love with the beautiful Mabel.

Later, the Pirate King, Frederic’s erstwhile captain, shows up to enforce the terms of his indenture, which (no one had realized) specify that Frederic is to serve not until his 21st year but until his 21st birthday. As the Pirate King points out (in song, natch):

For some ridiculous reason, to which however I’ve no desire to be disloyal
Some person in authority — I don’t know who, very likely the Astronomer Royal –
Has decided that, although for such a beastly month as February, twenty-eight days as a rule are plenty
One year in every four, his days shall be reckoned as nine and twenty
Through some singular coincidence (I shouldn’t be surprised if it were owing to the agency of an ill-natured fairy)
You are the victim of this clumsy arrangement, having been born in leap year on the twenty-ninth of February
And so, by a simple arithmetical process, you will easily discover
That, though you’ve lived twenty-one years, yet if we go by birthdays, you’re only five and a little bit over.

Frederic, an admitted “slave of duty,” reluctantly agrees that he is still bound to the pirates. Breaking the news to Mabel, he sings to her,

In 1940 I of age shall be,
I’ll then return and claim you, I declare it!

From that you might conclude that, if his twenty-first birthday is in 1940, then he’ll be 84 that year (21×4) and that he was therefore born in 1856, but you’d be wrong!

Everyone knows that leap year comes once every four years, but not as many people know that it’s more complicated than that. It’s a leap year if:

  • The year is a multiple of four;
  • But not if the year is a multiple of one hundred;
  • Except when the year is a multiple of four hundred.

1996 was a multiple of four, and not a multiple of 100, and not a multiple of 400, so it was a leap year. 2000 was a multiple of four, so it would have been a leap year, except that it was also a multiple of 100, so it would not have been a leap year, except that it was also a multiple of 400, so it was a leap year — and because of that, it allowed some people who believed “every four years” is the complete rule to continue believing that.

So by these rules, 1900 was not a leap year. If Frederic had been born in 1856, then 1940 would have been only his twentieth birthday, not his twenty-first. For 1940 to be his twenty-first birthday he had to have been born in 1852.

So the action at the beginning of The Pirates of Penzance takes place on 1 March 1873 — the first day after Frederic’s twenty-first year. (Assuming Frederic was aware that 1900 would not be a leap year when he told Mabel “1940″ — which, being such a stickler for the rules, he surely was.)

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