Here’s a tip for your next craps game. You can find the odds of rolling any number with two dice by subtracting the number from 7, ignoring the sign, and subtracting the result from 6. The remainder is the number of chances out of 36 that the number will appear.
For example, there are (6 – (7 – 5)) = 4 chances in 36, or 1 chance in 9, that you’ll roll a 5.
We’re told that, in any right triangle, a2 + b2 = c2. But consider:
In the figure above, the total length of the red line is 2(a/2) + 2(b/2), or a + b. And again:
Here the red line’s length is 4(a/4) + 4(b/4), which is still a + b.
With each iteration, the red line more closely approximates c, but its length remains a + b. At the limit, then, it seems, a + b = c. Was Pythagoras mistaken?
Here’s a curious way to multiply two numbers. Suppose we want to multiply 97 by 23. Write each at the head of a column. Now halve the first number successively, discarding remainders, until you reach 1, and double the second number correspondingly in its own column:
Cross out each row that has an even number in the left column, and add the numbers that remain in the second column:
That gives the right answer (97 × 23 = 2231). Why does it work?
This is rather amazing. Arrange a deck of cards in this order, top to bottom:
A♣, 8♥, 5♠, 4♦, J♣, 2♥, 9♠, 3♦, 7♣, Q♥, K♠, 6♦, 10♣,
A♥, 8♠, 5♦, 4♣, J♥, 2♠, 9♦, 3♣, 7♥, Q♠, K♦, 6♣, 10♥,
A♠, 8♦, 5♣, 4♥, J♠, 2♦, 9♣, 3♥, 7♠, Q♦, K♣, 6♥, 10♠,
A♦, 8♣, 5♥, 4♠, J♦, 2♣, 9♥, 3♠, 7♦, Q♣, K♥, 6♠, 10♦
Now:
Despite all this, you’ll find that the resulting deck is made up of 13 successive quartets of four suits–and four consecutive straights, ace through king.
The reasons for this are fairly complex, so I’ll just call it magic. You’ll find a full analysis in Julian Havil’s Impossible? Surprising Solutions to Counterintuitive Conundrums (2008).
Three condemned prisoners share a cell. A guard arrives and tells them that one has been pardoned.
“Which is it?” they ask.
“I can’t tell you that,” says the guard. “I can’t tell a prisoner his own fate.”
Prisoner A takes the guard aside. “Look,” he says. “Of the three of us, only one has been pardoned. That means that one of my cellmates is still sure to die. Give me his name. That way you’re not telling me my own fate, and you’re not identifying the pardoned man.”
The guard thinks about this and says, “Prisoner B is sure to die.”
Prisoner A rejoices that his own chance of survival has improved from 1/3 to 1/2. But how is this possible? The guard has given him no new information. Has he?
(In Mathematical Ideas in Biology [1968], J. Maynard Smith writes, “This should be called the Serbelloni problem since it nearly wrecked a conference on theoretical biology at the villa Serbelloni in the summer of 1966.”)
Suspend a metal coat hanger on a length of string, wrap the ends of the string around your index fingers, and insert your fingers in your ears. Now swing the hanger gently against the edge of table. What do you hear?
If the Earth did move at a tremendous speed, how could we keep a grip on it with our feet? We could walk only very, very slowly; and should find it slipping rapidly under our footsteps. Then, which way is it turning? If we walked in the direction of its tremendous speed, it would push us on terribly rapidly. But if we tried to walk against its revolving–? Either way we should be terribly giddy, and our digestive processes impossible.
— Margaret Missen, The Sun Goes Round the Earth, quoted in Patrick Moore, Can You Speak Venusian?, 1972
In 1957 ornithologists marked a Manx shearwater, a migrating seabird, on Bardsey Island off Wales.
In April 2002 they discovered the same bird was still alive and still gamely flying to South America each winter.
In the intervening 45 years, they calculated, it had covered 5 million miles.
See also Longest Migration.
