## Math Puzzles with Letters

*June 1, 2021 at 6:36 am* *
4 comments *

This week on the NPR Sunday Puzzle, host Will Shortz offered the following challenge:

Name a famous city in ten letters that contains an

s. Drop thes. Then assign the remaining nine letters their standard value in the alphabet — A = 1, B = 2, C = 3, etc. The total value of the nine letters is only 25. What city is it?

It’s not much of a spoiler to note that the average value of those nine letters must be less than three, since their sum “is only 25.” Consequently, a lot of those letters must occur at the beginning of the alphabet and — if eight of them were *a*‘s — there would be no letters later than *q* in the name of the city. But that’s as much as I’ll say; you can solve the puzzle on your own. (When you do, you can submit your answer for a chance to play next week’s on-air puzzle live with Will Shortz.)

Mathematician Harold Reiter uses a similar problem with elementary school students. Using the same idea — that each letter has a value (in cents) equal to its position in the alphabet — he asks students to find a dollar word, that is, a word whose letters have a sum of 100. As it turns out, there are many. Based on a nonexhaustive search, there are at least 3,500 dollar words, and likely a whole lot more. In a quick perusal of the list, one word jumped out: **oxygon**. Nope, that’s not a typo. It’s an archaic term meaning “a triangle with three acute angles.”

All of this talk of letters reminds me of my favorite puzzle, which I call Product Values. Using the same scheme — that is, A = 1, B = 2, C = 3, etc. — find the product value of a word by *multiplying* the values of the letters. So, for instance, *cat* has a product value of 3 × 1 × 20 = 60. How many words can you find that have a product value of 100? Based on the ENABLE word list, there are nine. (If you need some help, you can use the Product Value Calculator at www.mathjokes4mathyfolks.com.)

To end this post, a few math jokes that involve letters:

And Satan sayeth, “Let’s put the alphabet in math.” Bwa-ha-ha-ha-ha.

Romans had no trouble with algebra, because X was always equal to 10.

Entry filed under: Uncategorized. Tags: alphabet, letter, NPR, puzzle, Will Shortz, word.

1.Xander Henderson | June 7, 2021 at 11:10 amWhat is really interesting about this is that it doesn’t take too much more work to go from assigning each letter a number to proving incompleteness. Give every symbol in mathematics a number, and assign to each statement a number which is the product of prime powers. The first symbol in the statement gives (for example) a factor of 2 to a power, the second gives a factor of 3 to a power, and so on. We can use this to assign a unique number to every possible statement. Then create a self-referential statement, and watch Hilbert’s head explode.

2.Xander Henderson | June 7, 2021 at 11:28 amRegarding the puzzle, the possible factorizations of 100 (and the corresponding letters of the alphabet) are given by

2 2 5 5 BBEE

2 2 25 BBY

2 10 5 BJE

4 5 5 DEE

20 5 TE

100 –

As A has a value of 1, we can also throw in as many As as we like. A few minutes of playing around gives me the following words (many of which are likely not valid, but I’ll put them out there anyway).

ABBY

BABY

JEB (a name, not on the ENABLE list)

DEE (the letter D)

DEAD

TEA

EAT

ETA (the Greek letter)

ATE

ET (I found this while checking to see if ETA was on the list)

3.venneblock | June 7, 2021 at 1:51 pmGood to hear from you, Xander! Love the incompleteness proof.

I believe that DEAD actually has a value of 80, not 100.

ENABLE does not include either ABBY or JEB; or, at least, it didn’t several years ago when I first solved this problem.

From your list — both the factorizations and then the corresponding word(s) — I believe you’re missing one possibility. The hint is that it’s a rather common word.

4.Xander Henderson | June 7, 2021 at 2:29 pmOh, abbey has an E in it, don’t it? Drat. And I somehow mentally transformed that second E into a D in DEE. You should see what I do with numbers—it is even worse. There are reasons that I don’t play these kinds of letter games very often—the dyslexia gets me every time.

OH! I forgot 4 times 25. Derp. Four quarters in a dollar. D’oh.

4 is D, 25 is Y, so… DAY.