John Horton Conway,
the man who invented the Game of life
made many more interesting math games.
One of them is called
A Powertrain
A powertrain is a way of
rewriting numbers
with their digits alternately raised as
a power to the previous digit
Take 45632. Its powertrain is
4 raised to 5 into 6 raised to 3 into 2
and it reduces to 4 4 2 3
6 8
Reapply the power train again and again
and the number will reduce to a single digit.
So naturally, convey wanted to know if
there were any indestructible numbers
that did not reduce to a single digit
under the power train.
He could find only one 2592
which turns into itself on applying the
powertrain
Later his friend Neil Sloane took up the
chase and uncovered a second
Sloane then said he was confident that
there are no other indestructible digits
People had searched for more and none
was found
I too tried
got bored
and stopped
Consider that for a moment
These two
numbers are so special
If you picturize the power train
as a number-annihilating train
which runs on the number line and apply
the power train conversion to every
natural number
and destroy the ones which reduce to a single digit
then after the train runs through the whole track
the only numbers that will be left will be these two.
In the book Alex's adventures in numberland
Alex Bellos says:
Conway's powertrain is such a lethal machine
that it annihilates every number in the
universe
apart from these two seemingly unrelated
fixed points in the never-ending expanse
of numbers
Bigger numbers die fairly quickly under
the powertrain
because a zero appears and the whole
thing reduces to a zero
This seemingly random big number
is so special due to this fact
On asking whether it had any real world
application
Sloane said: It is just amusing
nothing wrong with that you have to have fun
