THE ALPHABET RACE

by Ross Eckler
Word Ways, 1999

Visualize the 26 letters of the alphabet lined up like horses at the starting gate of an immensely-long racetrack. One minute after the start, letters O, N and E each advance one mile but the remaining 23 have not budged. During the second minute, T, W and O each advance one mile, placing O in the lead at two miles and W, E and T tied for second at one mile each. During the third minute, E advances two miles, while H, T and R advance one mile, placing E in the lead at three miles.

Since there are one thousand vigintillion (minus one) number names, this promises to be a long race indeed. Which letter will be the eventual winner? When does the lead change as the race progresses? The table at the end shows the development of the race for the first 100 minutes, focusing on the four leading letters T, I, N and E

O is a fast starter but fades rapidly; after 39 minutes it has gone only 39 miles. Although E holds the lead at 100 minutes, T actually led the race more than half the time, having taken the lead at the end of 39 minutes and relinquished it at the end of 85 minutes. D, a new letter joins the race at 100 minutes. A steady performer, it advances two miles every minute, but this is not enough to overtake the leaders; E, for example, adds one mile per minute to its previous average speed of 1.48 miles per minute.

Will E ever be overtaken? One might bet on I when the millions are reached, but this is not enough to matter, as shown by Frank Rubin in the Nov 1981 Colloquy. MILLION through QUINTILLION have no E's and two or more I's, enough to let I slowly gain, but SEXTILLION adds an E enabling it to pull away. Dan Hoey has written a computer program to find that E leads until the 1,908,414,049,538,005,261th minute when N passes it for the first time. E and N exchange the lead a total of 6 times in the quintillions, but E regains the lead in the sextillions, not to be bettered until the nonillions when 27 more exchanges take place. However, E has the last word, regaining the lead for good in the decillions and holding it the rest of the way.

Beyond vigintillion, one must create a number nomenclature in order to continue. In the Book of Numbers (Springer-Verlag, 1996), Conway and Guy have done just that. At 1063 there are three I-E exchanges, ending with I in the lead until 10240 , after which a number of I-N exchanges occur. But is anyone still interested?

```      T    I    N    E		     T    I    N    E
1              1    1		51   74   32   38   67
2    1         1    1		52   76   33   38   67
3    2         1    3		53   78   34   38   69
4    2         1    3		54   79   35   38   69
5    2    1    1    4		55   80   37   38   70
6    2    2    1    4		56   81   39   38   70
7    2    2    2    6		57   82   40   39   72
8    3    3    2    7		58   83   42   39   73
9    3    4    4    8		59   85   44   41   74
10    4    4    5    9		60   86   45   41   74
11    4    4    6   12		61   87   46   42   75
12    5    4    6   14		62   89   47   42   75
13    7    5    7   16		63   91   48   42   77
14    8    5    8   18		64   92   49   42   77
15    9    6    9   20		65   93   51   42   78
16   10    7   10   22		66   94   53   42   78
17   11    7   12   26		67   95   54   43   80
18   12    8   13   29		68   97   56   43   81
19   13    9   15   32		69   98   58   45   82
20   15    9   16   33		70   99   58   46   84
21   17    9   18   35		71  100   58   48   87
22   20    9   19   36		72  102   58   49   89
23   23    9   21   40		73  104   58   50   93
24   25    9   21   40		74  105   58   51   95
25   27   10   22   42		75  106   59   52   98
26   29   11   23   43		76  107   60   53  100
27   31   11   25   46		77  108   60   55  104
28   34   12   26   48		78  110   61   56  107
29   36   13   29   50		79  111   62   59  110
30   38   14   29   50		80  112   63   59  111
31   40   15   30   51		81  113   64   60  113
32   43   16   30   51		82  115   65   60  114
33   46   17   30   53		83  117   66   60  117
34   48   18   30   53		84  118   67   60  118
35   50   20   30   54		85  119   69   60  120
36   52   22   30   54		86  120   71   60  121
37   54   23   31   56		87  121   72   61  124
38   57   25   31   57		88  123   74   61  126
39   59   27   33   58		89  124   76   63  128
40   60   27   33   58		90  125   77   65  129
41   61   27   34   59		91  126   78   68  131
42   63   27   34   59		92  128   79   70  132
43   65   27   34   61		93  130   80   72  135
44   66   27   34   61		94  131   81   74  136
45   67   27   34   62		95  132   83   76  138
46   68   28   34   62		96  133   85   78  139
47   69   28   35   64		97  134   86   81  142
48   71   29   35   65		98  136   88   83  144
49   72   30   37   66		99  137   90   87  146
50   73   31   37   66	       100  137   90   89  148
```