MathDB

Let

r>1

be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at

0

and a blue bead at

1

. In a move, Alice chooses one of the beads and an integer

k \in \mathbb{Z}

. If the chosen bead is at

x

, and the other bead is at

y

, then the bead at

x

is moved to the point

x'

satisfying

x'-y=r^k(x-y)

.
Find all

r

for which Alice can move the red bead to

1

in at most

2021

moves.

Problem Statement