Free Letters to a Young Mathematician by Ian Stewart Page A

the other letters unchanged; at some other stage, the vowel in the third position has to change to a consonant. Maybe other vowels and consonants wander in and out, too, but whatever else happens, we can now be certain that a vowel cannot change position in one step.
    How does the number of vowels in the word change? Well, it can stay the same; it can increase by 1 (when a consonant changes to a vowel), or it can decrease by 1 (when a vowel changes to a consonant). There are noother possibilities. The number of vowels starts at 1 with SHIP and ends at 1 with DOCK, but it can’t be 1 at every step, because then the unique vowel would have to stay in the same place, position three, and we know that it has to end up in position two.
    Idea: think about the earliest step at which the number of vowels changes. The number of vowels must have been 1 at all times before that step. Therefore it changes from 1 to something else. The only possibilities are 0 and 2, because the number either increases or decreases by 1.
    Could it be 0? No, because that means the word would have no vowels at all, and by definition no “word”
in our restricted sense can be like that. Therefore the word contains two vowels; end of proof. We’ve barely started analyzing the problem, and a proof has popped out of its own accord. This often happens when you follow the line of least resistance. Mind you, things really start to get interesting when the line of least resistance leads precisely nowhere.
    It’s always a good idea to check a proof on examples, because that way you often spot logical mistakes. Let’s count the vowels, then:
    SHIP      1 vowel
    SHOP      1 vowel
    SHOT      1 vowel
    SLOT      1 vowel
    SOOT      2 vowels
    LOOT      2 vowels
    LOOK      2 vowels
    LOCK      1 vowel
    DOCK      1 vowel
    The proof says to find the first word where this number is not 1, and that’s the word SOOT, which has two vowels. So the proof checks out in this example. Moreover, the number of vowels does indeed change by at most 1 at each step. Those facts alone do not mean that the proof is correct, however; to be sure of its correctness you have to check the chain of logic and make sure that each link is unbroken. I’ll leave you to convince yourself that this is the case.
    Notice the difference here between intuition and proof. Intuition tells us that the single vowel in SHIP can’t hop around to a different position unless a new vowel appears somewhere. But this intuition doesn’t constitute a proof. The proof emerges only when we try to pin the intuition down: yes, the number of vowels changes, but when ? What must the change look like?
    Not only do we become certain that two vowels must appear, we understand why this is inevitable. And we get additional information free of charge.
    If a letter can sometimes be a vowel and sometimes a consonant, then this particular proof breaks down. For instance, with three-letter words there is a sequence:
    SPA
    SPY
    SAY
    SAD
    If we count Y as a vowel in SPY but as a consonant in SAY (which is defensible but also debatable), then each word has a single vowel, but the vowel position moves. I don’t think this effect can cause trouble when changing SHIP into DOCK, but that depends on a much closer analysis of the actual words in the dictionary. The real world can be messy.
    Word puzzles are fun (try changing ORDER into CHAOS). This particular puzzle also teaches us something about proofs and logic. And about the idealizations that are often involved when we use math to model the real world.
    There are two big issues about proof. The one that mathematicians worry about is, what is a proof? The rest of the world has a different concern: why do we need them?
    Let me take those questions in reverse order: one now, and the other in a later letter.
    I’ve begun to observe that when people ask why something is necessary, it is usually because they feel uncomfortable doing it and are