by Burkard Polster and Marty Ross
The Age, 1 June 2009
The other day, one of us managed to catch a taxi numbered 6174. That may
appear like just another number, but it is in fact very special.
Starting
with 6174, we use the digits to make two new numbers. The first is 7641, with
the digits placed in descending order. The second number is 1467, the reverse of
the first one. Now subtract the second from the first and you get 7641 – 1467 =
6174, the number we started with. Pretty neat.
But there is more. Take a
different four-digit number, say 8208. Following the same recipe, you arrive at
8820 – 0288 = 8532. Apply the recipe again and you get 8532 – 2358 = 6174, which
again is our taxi number.
Does this always work? Obviously not for a
number such as 3333, where all the digits are the same: we immediately arrive at
0. And, applying the recipe to a number such as 1112 gives 999, and then 0 at
the next stage. There are 77 such exceptions, all resulting in 0 after one or
two steps. But every other four-digit number eventually arrives at
6174.
Mathematicians call 6174 Kaprekar’s constant, after D. R. Kaprekar,
the Indian mathematician who discovered its amazing property.
As it
happens, we are not the only mathematicians who travel by taxi. There is a
famous taxi story involving the great mathematicians G. H. Hardy and Srinivasa
Ramanujan. Hardy complained that the number of his taxi, 1729, was very boring.
But Ramanujan disagreed, immediately recognizing that 1729 is the smallest whole
number that can be written as the sum of two cubes in two different ways:
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