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