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Canadian Open Math Challenge
2018 Canadian Open Math Challenge
C2
C2
Part of
2018 Canadian Open Math Challenge
Problems
(1)
2018 COMC C2
Source:
12/6/2018
Source: 2018 Canadian Open Math Challenge Part C Problem 2 —--Alice has two boxes
A
A
A
and
B
B
B
. Initially box
a
a
a
contains
n
n
n
coins and box
B
B
B
is empty. On each turn, she may either move a coin from box
a
a
a
to box
B
B
B
, or remove
k
k
k
coins from box
A
A
A
, where
k
k
k
is the current number of coins in box
B
B
B
. She wins when box
A
A
A
is empty.
(a)
\text{(a)}
(a)
If initially box
A
A
A
contains 6 coins, show that Alice can win in 4 turns.
(b)
\text{(b)}
(b)
If initially box
A
A
A
contains 31 coins, show that Alice cannot win in 10 turns.
(c)
\text{(c)}
(c)
What is the minimum number of turns needed for Alice to win if box
A
A
A
initially contains 2018 coins?
Comc
2018 COMC