What’s in a Name?
The product value of a word can be calculated as follows:
Assign each letter of the alphabet a value as follows: A = 1, B = 2, C = 3, and so on. The product value of a word is the product of its letters. For instance, the word CAT has a product value of 60 because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.
During a recent webinar, I introduced participants to my collection of Product Value Puzzles. The following product value puzzle is credited to John Horton Conway:
Find an English word with a product value of 3,000,000.
Finding the solution is up to you. But I will give you some good news — there’s not a unique answer. In fact, there are two English words that satisfy the conditions of the problem.
What most folks found interesting, though, are the Product Value Calculators on my web site. With these two tools, you can:
- Enter an integer value, and the first calculator will return all words in the English language whose product value equals the number you enter.
- Enter a word, and the second calculator will return the product value.
One of the participants during the webinar said that her middle school students, when confronted with any type of math puzzle involving words, will first apply the rules of the puzzle to their name. Apparently, I’m not much different from a middle school kid, because that’s what I did, too. Turns out, my name has a product value of 1,710,720:
Patrick = 16 × 1 × 20 × 18 × 9 × 3 × 11 = 1,710,720
So, then I wondered, “Are there any other words that have a product value of 1,710,720?” Of course, I could have used the Product Value Calculators to find the answer, but that would have been unsatisfying. With a little trial-and-error, I found that blackboard also has a product value of 1,710,720:
blackboard = 2 × 12 × 1 × 3 × 11 × 2 × 15 × 1 × 18 × 4 = 1,710,720
There were three things about solving this problem that I really enjoyed:
- My strategy involved substitutions: I replaced a letter or a pairs of letters by other pairs of letters that have the same product value. For instance, the t and c in Patrick could be replaced by o and d, because both pairs have a product value of 60.
- Calculating the product values for Patrick and blackboard reveal two distinct factorizations for 1,710,720.
- How cool is it that I’m a mathy folk, and my name and blackboard have the same product value?
(Incidentally, my boss David found that his name and the word chalk have the same product value. Some would argue that its numerological destiny that we work together and are friends.)
So now I’ll offer the challenge to you. Can you find a word that has the same product value as your name? Good luck!
Of course, if that’s more thinking than you care to do right now, you could just access the product value calculator. But what fun would that be?