Professor Stewart's Hoard of Mathematical Treasures Read Online Free PDF Page A

writing, the favoured answer is that, as the cat nears the ground, some kind of antigravity effect kicks in, and the cat hovers just off the ground while spinning madly over and over.
    However, this argument has some logical loopholes, and it ignores basic mechanics. We’ve just seen that the mathematics of falling cats, and falling toast, lends scientific support to both adages. So what does the same mathematics say about a buttered cat?
    What happens depends on how massive the toast is compared with the cat. If the toast is an ordinary slice, the cat has no difficulty in coping with the small amount of extra angular momentum that the toast contributes, and still lands on its feet. The toast doesn’t land at all.
    However, if the toast is made of some kind of incredibly dense bread, 5 so that its mass is much larger than that of the cat, then Matthews’s analysis applies and the toast lands buttered-side down with the cat upside down waving its paws frantically in the air.
    What happens for intermediate masses? The simplest possibility is that there is a critical cat-to-toast mass ratio [C : T] crit , below which the toast wins and above which the cat wins. But it wouldn’t surprise me to find a range of mass ratios for which the cat lands on its side or, indeed, exhibits more complex transitional behaviour. Chaos cannot be ruled out, as any cat owner knows.

Lincoln’s Dog
    Abraham Lincoln once asked: ‘How many legs will a dog have if you call its tail a leg?’
    OK, how many?
     
    Discussion on page 281

Whodunni’s Dice
    Grumpelina, the Great Whodunni’s beautiful assistant, placed a blindfold over the eyes of the famous stage magician. A member of the audience then rolled three dice.
    ‘Multiply the number on the first dice by 2 and add 5,’ said Whodunni. ‘Then multiply the result by 5 and add the number on the second dice. Finally, multiply the result by 10 and add the number on the third dice.’
    As he spoke, Grumpelina chalked up the sums on a blackboard which was turned to face the audience so that Whodunni could not have seen it, even if the blindfold had been transparent.
    ‘What do you get?’ Whodunni asked.
    ‘Seven hundred and sixty-three,’ said Grumpelina.
    Whodunni made strange passes in the air. ‘Then the dice were—’
    What? And how did he do it?
     
    Answer on page 282

A Flexible Polyhedron
    A polyhedron is a solid whose faces are polygons. It has been known since 1813 that a convex polyhedron (one with no indentations) is rigid: it cannot flex without changing the shapes of its faces. This was proved by Augustin-Louis Cauchy. For a long time, no one could decide whether a non-convex polyhedron must also be rigid, but in 1977 Robert Connelly discovered a flexible polyhedron with 18 faces. His construction was gradually simplified by various mathematicians, and Klaus Steffen improved it to a flexible polyhedron with 14 triangular faces. This is known to be the smallest possible number of triangular faces in a flexible polyhedron. You can watch it flex on:
    demonstrations.wolfram.com/SteffensFlexiblePolyhedron/uk.youtube.com/watch?v=OH2kg8zjcqk
    You can make one by cutting the diagram from thin card, folding it, and joining the edges marked with the same letters. You can add flaps to do this, or use sticky tape. The dark lines show ‘hill’ folds, the grey ones ‘valley’ folds.

    Cut out and fold: dark lines are ‘hill’ folds, grey lines are ‘valley’ folds.

    Join the edges as marked to get Steffen’s flexible polyhedron.

But What About Concertinas?
    Hang on a mo - isn’t there an obvious way to make a flexible polyhedron? What about the bellows used by blacksmiths to blow air into a fire? Or for that matter, what about a concertina? That has a flexible series of zigzag flaps. If you replace the two big pieces on the ends by flat-sided boxes, which they almost are anyway, then it’s a polyhedron. And it’s flexible. So what’s the big deal?

    Although a concertina is