amfladmdLv1

13 Apr 2022

(R6) A player is declared the winner if the other player loses, as described in (R3).
For example, if the game begins with 10 beans in one jar and 10 beans in the other jar,
the sequence of play could be:
Turn Number 1 2 3 4 5 6 7
Number of beans removed by Franco 1 3 4 1
Number of beans removed by Sarah 2 5 2
Number of beans remaining in the jars 10, 9 10, 7 7, 7 7, 2 3, 2 1, 2 0, 2
On the next turn, Sarah cannot remove 5 beans since the greatest number of beans
remaining in either jar is 2 and so after exactly 7 turns, Sarah loses and Franco wins.

(a) At the beginning of the first game, there are 40 beans in one jar and 0 beans in
the other jar. After a total of 10 turns (5 turns for each of Franco and Sarah),
what is the total number of beans left in the two jars?
(b) At the beginning of the second game, there are 384 beans in one jar and 0 beans
in the other jar. The game ends with a winner after a total of exactly n turns.
What is the value of n?
(c) At the beginning of the third game, there are 17 beans in one jar and 6 beans in
the other jar. There is a winning strategy that one player can follow to guarantee
that they are the winner. Determine which player has a winning strategy and
describe this strategy. (A winning strategy is a way for a player to choose a jar
on each turn so that they win no matter the choices of the other player.)
(d) At the beginning of the fourth game, there are 2023 beans in one jar and
2022 beans in the other jar. Determine which player has a winning strategy
and describe this strategy.