Strange Math

Two problems that will make you want to throw a chair at someone, from John Jackson, Rational Amusements for Winter Evenings, 1821:


If from six ye take nine, and from nine ye take ten
(Ye youths, now the mystery explain;)
And if fifty from forty be taken, there then,
Shall just half a dozen remain.


One third of twelve, if you divide,
By just one fifth of seven,
The true result (it has been tried,)
Exactly is eleven.

How? Why?

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Lightning Addition

A (probably apocryphal) story tells that, as a 10-year-old schoolboy, Carl Friedrich Gauss was asked to find the sum of the first 100 integers. The tyrannical schoolmaster, who had intended this task to occupy the boy for some time, was astonished when Gauss presented the correct answer, 5050, almost immediately.

How did Gauss find it?

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There Goes the Neighborhood

A Martian sand lizard can reproduce itself in a single day. Start with a single sand lizard and on succeeding days you’ll have 2, then 4, and so on. In 30 days you’ll have 536,870,912 lizards.

How long would it take to reach that number if you started with two lizards?

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steadfast chess problem

By W. Bone. White to move and mate in four.

The catch: He must mate with the queen — and she’s glued to the board.

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Measuring the River

measuring the river

A traveler reaches a river at the point A and wishes to know the width across to B. As he has no means of crossing the river, what is the easiest way of finding its width?

From Henry Dudeney.

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Fire Escape

You’re new to hell, and you’re given a choice: You can go directly to the fourth circle, or you can play simultaneous chess games against Alexander Alekhine and Aron Nimzowitsch. Alekhine always plays black and smokes a pipe of brimstone. Nimzowitsch plays white and wears cufflinks made of human teeth. Neither has ever lost.

If you can manage even a draw against either player, you’ll be set free. But if they both beat you, you’ll go to the eighth circle for eternity.

What should you do?

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“Who Can Read Franklin’s Cipher?”

Benjamin Franklin wrote from Passy, in 1781, a letter to M. Dumas. He said:— ‘I have just received a 14, 5, 3, 10, 28, 2, 76, 203, 66, 11, 12, 273, 50, 14, joining 76, 5, 42, 45, 16, 15, 424, 235, 19, 20, 69, 580, 11, 150, 27, 56, 35, 104, 652, 20, 675, 85, 79, 50, 63, 44, 22, 219, 17, 60, 29, 147, 136, 41, but this is not likely to afford 202, 55, 580, 10, 227, 613, 176, 373, 309, 4, 108, 40, 19, 97, 309, 17, 35, 90, 201, 100, 677.’ This has never been deciphered. The state department at Washington has no key to it. I submit it for the consideration of the whole world.

— Elliott Sandford, New York World, cited in Henry Williams, A Book of Curious Facts, 1903

04/09/2014 Now solved!

12/30/2014 Oops, that link seems to have gone bad. Here’s another one. The numbers are keyed to the text of a book that Franklin’s correspondent Charles Dumas had sent to him. The message reads, “I have just received a neuu comiissjon joining me uuith m adams in negodiaions for peace but this is not likely to afford me much employ at present.”