# A Christmas Carroll

Posted by Graham Wheeler on Tuesday, March 29, 2011

On Christmas day, 1877, Lewis Carroll, author of the Alice books, entertained two bored young girls by inventing a new game that he called Word Links: given two words, change one word to the other by changing a single letter at a time with the intermediate steps all being valid words themselves. For example, to change “cold” to “warm”, one can use the steps “cord”, “card”, “ward”. Carroll later popularised this form of puzzle in a series of articles in Vanity Fair magazine, changing the name to Doublets - from the “double, double, toil and trouble” witches' incantation in Shakespeare’s Macbeth.

These puzzles are still popular today (usually called Word Ladders). The noted Stanford computer sciencist Don Knuth dedicated several pages of his work on graph algorithms (The Stanford Graphbase) to the topic. Knuth investigated five-letter words and found that of a set of 5,757 common English words, there were 14,135 links between words (where a link is a single ladder step). Only 671 words had no links - Knuth called these words ‘aloof’ - ‘aloof’ itself being aloof, along with words such as ’earth’, ‘ocean’, ‘sugar’ and ’laugh’. 103 pairs such as ‘opium’ and ‘odium’ exist. The two biggest connected sets have 25 words each. By relaxing the rules - for example by allowing the letters to be rearranged at each step - the words become considerably more connected.

Ted Johnson conducted an analysis of four letter words, with the additional rule of allowing the word to be reversed at any step. Starting with an online dictionary of 4776 words, he found that there was one huge linked component of 4436 words. This is not surprising; in 1960 Paul Erdős and Alfred Rényi proved that if the average number of links is sufficiently high thenthe set of words will form one large connected component with a few outliers. Word ladders are interesting because of their similarity to genetic mutation in DNA. Carroll himself came up with the evolutionary chain ‘ape’, ‘are’, ’ere’, ’err’, ’ear’, ‘mar’, ‘man’, although he was a sceptic of Darwin’s theories.

Consider the subsequence from ’err’ to ‘man’ - can you prove that to change one word with a single vowel to another word with a single vowel but in a different position, that it is necessary to go through an intermediate step of a word with two vowels? For the purpose of this problem, consider ‘y’ to be a vowel.