Self-expressive number names

Word Ways, Nov, 2007 by a. Ross Eckler

If one sets A=1, B=2, ... Z=26, it is well-known that no number-name is self-expressive (15 14 5 does not add to ONE, 20 23 15 to TWO, etc.). However, there are other ways to arithmetically combine numbers. If subtraction as well as addition is allowed, then five of the first twenty number-names can express themselves:

SEVEN -19 5 22-5 14 = 36-29 = 7

THIRTEEN -20 8 9-18 20-5 5 14 = 56-43 = 13

FOURTEEN 6 15-21 18 20-5-5-14 = 59-45 = 14

FIFTEEN -6-9 6 20-5-5 14 = 40-25 = 15

SEVENTEEN - 19-5-22 5 14 20 5 5 14 = 63-46 = 17

One can similarly self-express half of the number-names between TWENTY and EIGHTY: 21-3-5-8-9, 30-2-4-6-7, 40-2-4-6-7, 50-2-4-6-7, 61-3-5-8-9. 70-2-4-6-7. However, EIGHTY cannot be self-expressed because its letters sum to only 74, and NINETY similarly fails because it sums to 87. NINETY-FIVE, despite having ten numbers to work with, fails just as ONE through SIX did; one cannot allocate plus and minus signs to 14,9,14,5,20,25,6,9,22,5 so that positive numbers add to 112 and negative ones to 17.

As number-names grow larger, self-expressive ones become scarcer for both of the above reasons. The largest self-expressive ones appear to be

TWO HUNDRED NINETY-SIX (58 109 87 52 = 306) 301-5(E) = 296

THREE HUNDRED TWENTY-TWO (56 109 107 58 = 330) 326-4(D) = 322

THREE HUNDRED TWENTY-FOUR (56 109 107 60 = 332) 328-4(D) = 324

One can generate additional self-expressive number-names by allowing multiplication and division. One can produce even more by allowing the square root and the factorial (for example, 4! = 4x3x2x1 = 24). However, the components must be used in order, a requirement that is irrelevant when only addition and subtraction is considered. Here are additional self-expressive numbers less than or equal to TWENTY:

THREE (20,8,18,5,5) 20/(-8 18) 5/5 = 3

EIGHT (5,9,7,8,20) 5 [square root of (9 7)] 8 - 20 = 8

TEN (20,5,14) -20/5 14 = 10

ELEVEN (5,12,5,23,5,14) -5 - 12/5 22/5 14 = 11

TWELVE (20,23,5,12,22,5) -20 - 23 5 (12-22)(-5) = 12

SIXTEEN (19,9,24,20,5,5,14) 19 9 - 24 - 20/(5 5) 14 = 16

EIGHTEEN (5,9,7,8,20,5,5,14) (5 9)/(-7) 8 -20/(5 5) 14 = 18

NINETEEN (14,9,14,5,20,5,5,14) 14 9 - 14 - 5 - (20/5) 5 14 = 19

TWENTY (20,23,5,14,20,25) -(-20 23)! - 5 - 14 20 25 = 20

Alas, most three-letter and four-letter number-names possess too few components to work with.

The largest self-expressive number-name using all the arithmetic operations (including factorial) is extraordinarily difficult to determine, requiring the use of a computer. Not counting the six single-digit number-names having only three or four letters, it should be somewhat easier to determine the smallest number-name that is not self-expressive.

I am indebted to Dave Morice for the concept of self-expressive numbers by arithmetic operation.

A. ROSS ECKLER

Morristown, New Jersey