Alex’s Adventures in Numberland (46 page)

Likewise, if we worked it out here, we would find that 277777788888899 has persistence 11. Yet here’s the thing: Sloane has never discovered a number that has a persistence greater than 11, even after checking every number all the way up to 10
²³³
, which is 1 followed by 233 zeros. In other words, whatever 233-digit number you choose, if you follow the steps of multiplying all the digits together according to the rules for persistence, you will get to a single-digit number in 11 steps or fewer.

This is splendidly counter-intuitive. It would seem to follow that if you have a number with 200 or so digits consisting of lots of high digits, say 8s and 9s, then the product of these individual digits would be sufficiently large that it would take well over 11 steps to reduce to a single digit. Large numbers, however, collapse under their own weight. This is because if a zero ever appears in the number, the product of all the digits is zero. If there are no zeros in the number to start with, a zero will
always
appear by the eleventh step, unless the number has already been reduced to a single digit by then. In persistence Sloane found a wonderfully efficient giant-killer.

Not stopping there, Sloane has compiled the sequence in which the
n
th term is the smallest number with persistence
n
. (We are considering only numbers with at least two digits.) The first such term is 10, since:

I find this list of numbers strangely fascinating. There is a distinct order to them, yet they also are a bit of an asymmetric jumble. Persistence is sort of like a sausage machine that produces only 11 very curiously shaped sausages.

Sloane’s good friend Princeton professor John Horton Conway also likes to amuse himself by coming up with offbeat mathematical concepts. In 2007 he invented the concept of a powertrain. For any number written
abcd
…, its powertrain is
a
^b
c
^d
…In the case of numbers where there is an odd number of digits, the last digit has no exponent, so
abcde
goes to
a
^b
c
^d
e
. Take 3462. It reduces to
³⁴
6
²
= 81 × 36 = 2916. Reapply the powertrain until only a single digit is left:

Conway wanted to know if there were any indestructible digits – numbers that
did not
reduce to a single digit under the powertrain. He could find only one: