MathDB

Source: 2018 Canadian Open Math Challenge Part C Problem 2 —--
Alice has two boxes

A

and

B

. Initially box

a

contains

n

coins and box

B

is empty. On each turn, she may either move a coin from box

a

to box

B

, or remove

k

coins from box

A

, where

k

is the current number of coins in box

B

. She wins when box

A

is empty.

\text{(a)}

If initially box

A

contains 6 coins, show that Alice can win in 4 turns.

\text{(b)}

If initially box

A

contains 31 coins, show that Alice cannot win in 10 turns.

\text{(c)}

What is the minimum number of turns needed for Alice to win if box

A

initially contains 2018 coins?

Problem Statement