# Packing Numbers

Leonard Gordon noted this interesting pattern in the May 1995 issue of Word Ways. The English names of the first eight positive integers (ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT) contain altogether 32 letters. The smallest rectangular grid into which they can all be packed, word-search fashion, is 5×5. Because some of the cells serve double duty, the 32 letters “fit” into 25 cells; the ratio of these values is 1.28. This ratio remains remarkably consistent as the list of numbers is extended — here are grids for the first 8, 9, 10, 11, and 12 numbers:

```E I G H T   O N E E R H T   E I G H T F   E L E V E N S   S E V E N O W T
F O U R W   S E V E N I N   F   X   W O   I   F S O I E   F I V E E R H T
X I S   O   E I G H T W O   N I N E O U   G   O I T N V   O G X N E N I N
S E V E N   F O U R X I S   S E V E N R   H W U X V E E   U H E V L E W T
T H R E E                   T H R E E     T H R E E E N   R T N E V E L E
8 words     9 words        10 words      11 words        12 words
32 letters  36 letters      39 letters    45 letters      51 letters
25 cells    28 cells        30 cells      35 cells        40 cells
(1.28)      (1.29)          (1.30)        (1.29)          (1.28)
```

Alas, the last one isn’t optimal, Gordon notes. The names ONE through TWELVE will fit into a more compact grid:

```T W E L V E
F N E X S L
O F I V E E
U S G N V V
R T H R E E
O W T E N N
```

… and that raises the ratio to 1.42 letters per cell.

(Leonard Gordon, “Packing the Cardinals,” Word Ways 28:2 [May 1995], 116.)