MathDB

For any integer

n\geq 2

, we compute the integer

h(n)

by applying the following procedure to its decimal representation. Let

r

be the rightmost digit of

n

. [*]If

r=0

, then the decimal representation of

h(n)

results from the decimal representation of

n

by removing this rightmost digit

0

. [*]If

1\leq r \leq 9

we split the decimal representation of

n

into a maximal right part

R

that solely consists of digits not less than

r

and into a left part

L

that either is empty or ends with a digit strictly smaller than

r

. Then the decimal representation of

h(n)

consists of the decimal representation of

L

, followed by two copies of the decimal representation of

R-1

. For instance, for the number

17,151,345,543

, we will have

L=17,151

R=345,543

and

h(n)=17,151,345,542,345,542

. Prove that, starting with an arbitrary integer

n\geq 2

, iterated application of

h

produces the integer

1

after finitely many steps.
Proposed by Gerhard Woeginger, Austria

Problem Statement