Professor Stewart's Hoard of Mathematical Treasures (52 page)

Well, the dog may have lost its tail, or some legs, or it might be a mutant with six legs and five tails ... Quibbles aside, this is a good question to distinguish mathematicians from politicians. Lincoln asked his question in the context of slavery, putting it to a political opponent who maintained that slavery was a form of protection, trying to imply that it was benign. Lincoln’s answer was: ‘It has four legs - calling a tail a leg doesn’t make it a leg.’ By which he meant that calling slavery protection didn’t make it protection, which in that context is fair enough. Barack Obama’s famous ‘lipstick on a pig’ remark made the same point, though his opponents chose to interpret it as an insult to Sarah Palin.
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Ignoring the context, though, most mathematicians would beg to differ with President Lincoln, and answer ‘five’. Renaming a tail as a leg amounts to a temporary redefinition of terminology, which is common in mathematics. For example, in algebra ‘the’ unknown is usually denoted x, but the value of x differs from one problem to the next. If x was 17 in last week’s homework it need not be 17 ever after. The usual convention is that a temporary redefinition remains in force until it is explicitly cancelled, or until the context makes clear that it has been cancelled.

In fact, mathematicians habitually go further, and permanently redefine important terminology, usually to make it more general. Concepts such as number, geometry, space and dimension are examples; their meaning has changed repeatedly as the subject has progressed.

So, to mathematicians, if we agree to call a tail a leg for the rest of the discussion - which Lincoln’s question tacitly assumes, otherwise it’s not worth asking - then the meaning of ‘leg’ has changed, and now includes tails. So, Mr President, the dog has five legs, by your own redefinition.

What happens to Lincoln’s political point? It remains intact, but for a different reason. When Lincoln’s opponent asserted that slavery is protection, he redefined protection for the remainder of the discussion, so the properties normally associated with protection might not apply any longer. In particular, the new meaning does not imply that slavery is an act of kindness.

The dice were 5, 1 and 3.

If the dice show the numbers a, b and c, then the calculation produces in turn the numbers

So Whodunni subtracted 250 from 763, to get 513 - the numbers on the three dice. Subtract 2 from the first digit of the answer, 5 from the second, and leave the third alone - easy.

Heron’s formula applies to a triangle with sides a, b, c, and area x. Let s be half the perimeter:
Then Heron proved that
Square this equation and rearrange to get
This is a polynomial equation relating the area x to the three sides a, b, c.

The other 3-digit numbers that are equal to the sum of the cubes of their digits are 370, 371, and 407.

If the digits are a, b, c, then we have to solve
with 0 ≤ a, b, c ≤ 9 and a > 0. That’s 900 possibilities, so a systematic search will find the answer.

The work can be reduced by using some fairly simple tricks. For instance, if you divide a perfect cube by 9, the remainder is 0, 1 or 8. If you divide 100 or 10 by 9, the remainder is 1. So
a
+
b
+
c
and
a
³
+
b
³
+
c
³
leave the same remainder on division by 9. Eliminating cases where the digits are too small or too big to work,
a
+
b
+
c
has to be one of 7, 8, 9, 10, 11, 16, 17, 18, 19, 20. After that ... well, you get the idea. It’s a bit of a scramble, but it can be pushed through. Maybe there’s a better way.

There are plenty of solutions (usually there are lots, or none). Here’s one for each puzzle:

If you’re worried about EIDER and SOOTS, the first is a kind of duck and the second is not the plural of ‘soot’ (which is ‘soot’) but derives from the verb ‘to soot’, which means to cover with soot. As in the lesser-known proverb ‘A clumsy chimney-sweep soots the hearth.’
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Both words are in the official Scrabble
^TM
dictionary.

Now, I promised some maths, and we’ve not seen any yet.

All these puzzles are really about networks (also called graphs), which are collections of dots joined by lines. The dots represent
objects, and the lines are connections between these. In the SHIP- DOCK puzzle, the objects are four-letter words, and two words are connected if they differ by just one letter (in a specific position). All four-letter word puzzles of this kind reduce to the same general question: Is the initial word connected to the final one by some path in the network of all possible four-letter words?

Connect SHIP to DOCK.

The diagram shows just a tiny part of this network - enough to find an answer.

There are computer algorithms (procedures) for finding paths between any two nodes of a network, and the mathematics quickly becomes fairly deep and difficult. One relatively simple point is that the whole network breaks up in to one or more components, and all the words in a component are connected to each other by paths. Once you have succeeded in joining a word to one of these components, you can easily join it to all the other words in that component.

How many components are there? A theorem proved by Paul Erdős and Alfred Rényi in 1960 implies that if on average each word connects to enough others - more than some critical amount - then we should expect to find one giant component containing almost all the words, and a scattering of much smaller ones. And this is what happens. The giant component usually misses some bits out - for
instance, if we can find an isolated word, one that has no immediate neighbours, then that word on its own would form one component, disconnected from everything else.

What about an obscure word like SCRY (meaning to crystal-gaze)? Is that isolated? No, SCRY connects to SPRY, then to SPAY, then SPAR, SPAN, ... and it has clearly ‘escaped’, with lots of potential links, so we expect it to join up to the giant component, even though my picture doesn’t show how. In fact, SPAY-SPAT- SPOT-SHOT will do. This is why there is probably only one giant component. Since it’s so big, anything that is linked by a path to a reasonable number of words has more and more potential links, and at some point the path will run into the giant component.

Ted Johnson analysed the network of four-letter words, with one slight change to the definition of a link: you are also allowed to reverse the word. This probably does not change the components significantly, if at all, because relatively few words are meaningful when reversed.

He obtained his list of four-letter words from an online dictionary, resulting in a total of 4,776. Using mathematical methods (the Graph module for the computer scripting language Perl) he found that some words are isolated (like HYMN, according to the Scrabble dictionary) or form isolated pairs. Another small component contains just eight words. That left 4,439 words: one giant component with 4,436 words, and a small one with three - TYUM, TIUM, TUUM. These are not in the Scrabble Dictionary, but tuum is a literary word for ‘yours’, from the Latin, as in ‘meum and tuum’ - mine and yours. I’m inclined to rule out the other two and count tuum as a single isolated word. His results can be found at:
users.rcn.com/ted.johnson/fourletter.htm

If you play around with the network, you start to notice some regular structural features. The group of words BAND, BEND, BIND, BOND is an example: they are all connected to each other. That’s because all the changes involve the same letter position, the second from the left. Biologists working on genetic networks call common small sub-networks motifs. There are five-word motifs like this, too: MARE, MERE, MIRE, MORE, MURE is one example.

A more significant motif in the word network is a series of three
words like SHOT-SOOT-SORT with two vowels in the middle word. Vowels are crucial. Most single-letter changes to words change a consonant to another consonant or a vowel to another vowel. If all changes were like that, the vowel positions could never move. So changing SHIP, with a vowel in position 3, to DOCK, with a vowel in position 2, would be impossible. But sometimes consonants can change to vowels, or vowels to consonants. A sequence like SHOT- SOOT-SORT in effect moves the location of the vowel, by introducing another one and then losing the first.

In going from ORDER to CHAOS, the biggest problem is moving the vowel positions around. That’s where EIDER and SOOTS come in, in fact. But notice that although both the start and end words have a vowel in position 4, some of the intermediate words don’t. Sometimes you have to take a detour to get where you want to go.

Provided we take a relaxed view of what constitutes a vowel, every English word contains one. The standard vowels are AEIOU, of course. But the Y in SPRY acts like a vowel, for instance, and Y is often included in the list of vowels. The same goes for the W in the Welsh word CWM (which comes into the 4-letter network if we use the plural CWMS). If we define vowels that way, or exclude words without vowels, then the Ship-Dock theorem holds. This states that when going from SHIP to DOCK, some intermediate word must contain two vowels.

Why? At each stage the number of vowels can change by at most 1, and if it does not change, then the vowel stays in the same position. If the vowel count was always 1, then the vowel in SHIP would have to stay in the third position - but the vowel in DOCK is in the second position. So the vowel count must change. Look at the first word where it changes. It starts at 1 and changes by 1, leading to either 2 or 0. But 0 is ruled out by our convention about what constitutes a vowel or a permissible word, so it must be 2. The same theorem holds for words of any length. If the initial word has a vowel where the final one has a consonant, or vice versa, then somewhere we must hit two or more vowels. Why more? Because of examples like ARISE-AROSE, where the start and end words have more than two vowels.