Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart Read Free Book Online Page B

intoxicated for increased gullibility.) The con-artist places three cups (or glasses) upright on the bar:
    He inverts the centre cup
and explains that he will now turn all three of them to the upside-down position in exactly three moves, where each move inverts exactly two cups. They need not be adjacent: any two will do. (Of course, this can be done in one move - invert the two end cups - but the requirement to use three moves is part of the misdirection.)
    The three moves are:
    Now the con-artist begins to work on the mug. He casually turns the middle cup upright to get
and invites the mug to repeat the trick, with a small bet on the side to make things more interesting.

    Strangely, the cups refuse to behave themselves, no matter what moves the mug attempts. What the mug fails to notice is that the initial position has been changed, surreptitiously. And even if he does notice the change, he may not be aware of the devastating consequences. The parity (odd/even) of the set of upright mugs has now changed from even to odd. But every move preserves this parity. The number of upright cups changes by -2, 2 or 0 at each move, so even numbers stay even and odd numbers stay odd. The first starting position has even parity, and so does the required final position. But the second initial position has odd parity. This makes the required final position inaccessible - not just in three moves, but in any number whatsoever.
    This disgraceful con-trick (please do not try this at home, or in a bar, or anywhere else - or if you do, keep me out of it) shows that there can be obstacles to cup-inversion, but it also misdirects the mug into looking for a three-move solution when the original problem can actually be solved in one move.
    The problem can be generalised, with a slight difference from the pub scenario. The resulting puzzle involves the same principles, but it’s tidier. I’ll ask you two instances of it.
Cups Puzzle 1
    Suppose you start with 11 cups, all upside down. The rule is that you must make a series of moves, each of which inverts precisely 4 cups. They do not have to be adjacent. Your objective is to get all 11 cups upright at the same time. Can you do this, and if so, what is the smallest number of moves that does the job?
Cups Puzzle 2
    The same question starting with 12 cups, all upside down. Now the rule is that each move must invert precisely 5 cups. Again they do not have to be adjacent. Your have to get all 12 cups upright at the same time. Can you do this, and if so, what is the smallest number of moves required?
     
    Answers on page 287

Secret Codes
    Coded messages are as old as writing, but most of the early codes were very easy to break. For instance, the message
    QJHT EP OPU IBWF XJOHT
can be decoded as
    PIGS DO NOT HAVE WINGS
just by changing each letter into the previous one in the alphabet. If the code shifts the entire alphabet along a number of steps, there are only 25 possibilities to try. Julius Caesar is thought to have used this kind of code, with a shift of 3, in his military campaigns. It has the advantage that encrypting messages (putting them into code) and decrypting them (working out the original ‘plaintext’ from the coded message) are easy. Its main disadvantage is that you don’t have to be very bright to break the code.
    You don’t have to keep the alphabet in (cyclic) order, of course; you could shuffle it into some random-looking order, giving a substitution code. Both sender and recipient must know the shuffled order, so they probably have it written down somewhere, which is potentially insecure. Or else they remember a ‘key’ such as DANGER! FLYING PIGS, which reminds them to use the order
    DANGERFLYIPSBCHJKMOQTUVWXZ
which starts with the letters of the key, ignoring duplicates, and finishes with all the others in alphabetical order. Or maybe reverse alphabetical order, if lots of letters happen to remain unchanged.
    Substitution codes are easy to break if the person trying to