ppp: The Sum of Its Parts

"I thought ... I might trick myself into ... more precise language and not go chasing across your pages, my dear Journal, like a wild-eyed Diana shooting in all directions in the hope of bringing down one perfect word."

—Dorcas, in Sesyle Joslin's The Young Savages

Before I can talk about this week's topic, I need to provide a little mathematical background.

A perfect number is defined as a number which is equal to the sum of all of its divisors (that is, the sum of all the numbers that go into it evenly). For instance, the divisors of 6 are 1, 2, and 3; 1 + 2 + 3 = 6, so 6 is a perfect number. The divisors of 8 are 1, 2, and 4; 1 + 2 + 4 = 7, not 8, so 8 is not a perfect number. The divisors of 28 are 1, 2, 4, 7, and 14; if you add them up, you get 28, so 28 is perfect. The first four perfect numbers are 6, 28, 496, and 8128. For a great deal of information about the history of perfect numbers, see this perfect numbers page. (And I would be remiss in my duties as an alumnus if I didn't mention that there's a page at Swarthmore College that lists the first twenty perfect numbers in full.)

So, I hear you ask, what does this have to do with words? Well, about a year and a half ago David Van Stone introduced me to his notion of a "perfect word." If you assign numeric values to letters (A = 1, B = 2, and so on), a perfect word is an English word in which the number of the highest-numbered letter is equal to the sum of the numbers of all the other letters in the word. For instance, in the words alm and lam, A (1) + L (12) = M (13). Similarly, A + E + F = L (that is, 1 + 5 + 6 = 12), so leaf is a perfect word; and hazel is a perfect word, because A + E + H + L = Z (1 + 5 + 8 + 12 = 26).

Coming up with 3- and 4-letter perfect words isn't too difficult; you can often find them by picking a combination of letters that adds up properly and then rearranging them until you get a word. But not many perfect words more than four letters long are known. Finding them can be a fascinating and frustrating process; near misses are easy to find, but adjusting near misses so that the letters add up generally results in strings of letters that aren't English words. For instance, you might start with the word mayday, hoping to derive a perfect word from it. Its highest-numbered letter is Y, and there are two Ys, so it can't be a perfect word itself (because Y plus other letters can't equal Y). But if you add up the rest of the letters, you find that A + A + D + M is S, 6 short of Y; so if you change one of the Ys in mayday to an F (6), you get a string of letters that adds up like a perfect word: A + A + D + F + M = Y. Unfortunately, madfay and mayfad aren't English words.

Sadie would be a perfect word, but I think as with most wordgames it's best to avoid proper names. Other strings of letters that add up properly but aren't words include acheddy, bugafe, cafeo, dafted, flexa, lagday, MacFax, salf, whadee, and xleaf.

(Note the preponderance of the letter A. A is quite useful in constructing perfect words, because its value is 1 and it's a vowel. E can also be useful, but perfect words using the other vowels—especially U—are rare.)

Of course any anagram of a perfect word is another perfect word. It's also sometimes possible to find a new perfect word by taking a known one and adjusting one letter up while adjusting another letter down. For instance, habit is perfect; if you subtract 3 from H and add 3 to I, then rearrange the letters, you get table, which is thus also perfect. Similarly, ideas becomes bakes by adding 2 to the I and subtracting 2 from the D (and then rearranging the letters). And of course you can use this approach starting from any perfect string (whether a word or not): just start adjusting letters up and down and try to rearrange them into a word. Remember, though, that if you change the sum of the other letters, you must also change the highest-valued letter to reflect the new sum.

One could easily write a computer program to go through a dictionary file and check all the words there for perfection. I think it's more interesting, and more challenging, to find them by hand. If you want more examples before you start looking for your own, here's a list of the perfect words I know about. Please send me any others you find. If you used a computer to find them, let me know that too.

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