Magazine article Word Ways

# Self-Expressive Number Names

Magazine article Word Ways

# Self-Expressive Number Names

## Article excerpt

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. …

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