Professor Stewart's Hoard of Mathematical Treasures (50 page)

Now keep going, obeying, for increasing n, this rule:

Got that? Here’s the question: how long can you keep this process going?

At first sight, the answer seems to be: as long as you wish. After all, you can divide the line into indefinitely ever finer pieces, and choose points in whichever of those are appropriate.

I really don’t expect you to get the correct answer here, but I don’t want to give it away immediately, so you’ll find it on page 336. It’s amusing to try placing the first five or six points. Even then, it’s not as easy as it sounds.

In Flatland (page 255) the world is a plane and its inhabitants are geometric shapes. Flatlanders play their own versions of Spaceland games, and one of them is chess. The Flatland chessboard is eight cells long, and each player has three pieces: king, knight and rook, starting in the position shown.

Start of a game of Flatland chess.

The rules resemble those of Spaceland chess, bearing in mind the limitations of Flatland geometry. All three pieces can move to the left or to the right, if a suitable space is available. All moves must end either on an empty cell, or on a cell occupied by an enemy piece, which is then removed from the board - ‘taken’.

If a player has no legal move available, the game is stalemate, and is a draw. If a player can threaten the opposing king, and the king cannot escape, that’s checkmate, and the game is won.

If White plays first, and both players adopt perfect strategy, who will win?

 

The Infinite Lottery involves infinitely many bags: one numbered 1, one numbered 2, one numbered 3, one numbered 4, and so on. Each bag contains infinitely many lottery balls with the corresponding number.

You are supplied with a large box. You may place any number of balls you like into this box, chosen from any of the bags. There is just one condition: the total number must be finite.

Now you are required to change the balls in the box. You must remove and discard one, and replace it by as many balls as you like that bear smaller numbers. For instance, if you discard a ball with the number 100 on it, you can add to the box 10 million balls with the number 99, 17 billion with the number 98, and so on. There is thus no upper limit to the number of balls that can replace that solitary number-100 ball.

You must keep doing this. At each stage you may replace the discarded ball by whichever combination of balls you wish, provided you make their number finite and make sure that they bear smaller numbers than the ball you discarded. If you remove a ball marked 1, you cannot replace it because there are no balls with smaller numbers on them.

If eventually you run out of balls and empty the box, you lose. If you keep removing balls for ever - that is, if you never run out of balls - then you win.

But can you win the infinite lottery? If so, how?

 

In the days when people crossed the Atlantic in passenger liners, a ship left London every day at 4.00 p.m. bound for New York, arriving exactly 7 days later.

Every day at the same instant (11.00 a.m. because of the time
difference) a ship left New York bound for London, arriving exactly 7 days later.

All ships followed the same route, deviating slightly to avoid collisions when they met.

How many ships from London does each ship sailing from New York encounter during its transatlantic voyage, not counting any that arrive at the dock just as they leave, or leave the dock just as they arrive?

 

Mathematicians often use a technique called proof by contradiction. The idea is that to prove some statement true, you begin by assuming it to be false, and go on to derive various logical consequences. If any of these consequences leads to a logical impossibility - a contradiction - then your assumption that the statement is false cannot be correct. Therefore the statement is true.

You may have come across this by the name used in Euclid, which in Latin translation is reductio ad absurdum - reduction to the absurd.

For example, to prove that pigs don’t have wings, you first assume that they do, and deduce that pigs can fly. But we know that they can’t, so this is a logical impossibility. Therefore it is false that pigs don’t have wings, so they do.

Got that?

I will now use proof by contradiction to show that the largest whole number is 42.

Let n be the largest whole number, and suppose for a contradiction that n is not 42. Then n > 42, so (
n
- 42)
³
>0, which expands to give

Adding n to each side,
But the left-hand side is a whole number. Since it is greater than n, which we are assuming to be the largest whole number, we have derived a contradiction.

Therefore it is false that the largest whole number is not 42. So it is 42!

Clearly something is wrong here - but what?

 

Wherein the perspicacious or perplexed reader may procure answers to those questions that are presently known to possess answers ... together with such gratuitous facts and fancies as may facilitate their further delectation and enlightenment.

Past: 1961; Future: 6009.

If you insist on allowing a squiggle on the 7, amend these to 2007 and 2117.