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Each hit adds 7 points to Shane’s score, and so 4 hits adds
For each miss, 3 points are subtracted from Shane’s score, and so 2
misses subtracts
A player’s score begins at 0, and so after these 6 throws, Shane’s score
is
Solution 1
After exactly
After these throws, Susan’s score is 59, and so
Solving this equation, we get
Solution 2
Susan’s 6 misses decrease her score by
Since her score is 59, then Susan must have scored
Since each hit is worth 7 points, then the value of
Solution 1
We begin by making an initial guess at the value of
If we begin with
Since this score is greater than 105, then the value of
When
Since this score is greater than 85 and less than 105, then 4 is a
possible value of
When
This score is also greater than 85 and less than 105, and so 5 is a
possible value of
When
This score is less than 85 and so 6 is not a possible value of
Continuing to increase the number of misses will further decrease
Souresh’s score, and thus the only possible values are
Solution 2
Since Souresh makes 20 throws and
After exactly
Since Souresh’s score (
Since
Since Souresh’s score is less than 105, then
Since
Thus,
(We can check that when
Solution 1
The area of
Thus, the area of
Solution 2
The area of
Since the area of
The area of
Let the area of the overlapped region,
The area of
Since
Thus, the area of the figure
Since
Since the area of
We use the notation
Since
The prime factorization of
The positive integer
Since
In other words,
Suppose
If
Thus,
Suppose that
Since
In other words,
To begin, suppose that
That is, suppose that
In this case,
Since
Using these restrictions, there are exactly three possibilities for
which
If
If
Since each of these values is less than 500, then there are 3 positive
integers that satisfy the given conditions, in this case.
Next, suppose that
That is, suppose that
If
In this case,
Increasing
If
For which prime numbers
Since
The prime numbers in this range are 5,7,11,13,17,19, and 23, which give
7 positive integers that satisfy the given conditions, in this case.
Finally, suppose that
That is, suppose that
If
Increasing
Thus, the number of positive integers less than 500 that have the
factors 2 and 9 and exactly ten other positive factors is
Since one of the jars contains 0 beans, then all beans must be
removed from the jar that contains 40 beans.
On Franco’s turns, the number of beans that he removes is 1, 3, 4, 1, 3,
4,
After Franco’s 5 turns, he has removed a total of
On Sarah’s turns, the number of beans that she removes is 2, 5, 2, 5, 2,
5,
After Sarah’s 5 turns, she has removed a total of
After a total of 10 turns, the total number of beans removed by Franco
and Sarah is less than 40, and so we know that each of the two players
was able to remove the required number of beans on each of their 5
turns.
After a total of 10 turns, the total number of beans left in the two
jars is
Since one of the jars contains 0 beans, then all beans must be
removed from the jar that contains 384 beans.
Franco repeatedly removes
Sarah repeatedly removes 2 and 5 beans, which is a cycle of length
2.
Since the lowest common multiple of 3 and 2 is 6, then after each player
has had 6 turns (12 turns in total), they will each be back at the start
of their cycle.
After the first 12 turns (6 turns for each player), Franco removes
From the start of the game, together they will remove
After 10 such groups of turns (a total of
After these 120 turns, Franco and Sarah are at the beginning of their
sequences and it is Franco’s turn.
On the 121
On the 122
On the 123
On the 124
The next turn is Franco’s and he is not able to remove the required
number of beans, 4, and thus Franco loses.
Therefore, Sarah wins after exactly
We use the notation T
The game begins with the distribution
After T4 (2 turns each), Franco has removed
Thus after T4, there are a total of
With 12 beans remaining, at least one of the jars must contain 6 or more
beans, since if both jars contained fewer than 6 beans, then the total
number of beans would be at most
Since at least one of the jars contains 6 or more beans, then on T5,
Franco is able to remove 4 beans leaving a total of
With 8 beans remaining, at least one of the jars must contain 4 or more
beans, and so on T6, Sarah is able to remove 2 beans, and on T7, Franco
is able to remove 1 bean.
After T7 (4 turns for Franco and 3 for Sarah), the total number of beans
remaining in the jars is
Sarah must remove 5 beans on T8 (Sarah’s 4
If the distribution of beans is
However, if the distribution is
We summarize the solution to this point:
It is not possible for either player to win on T1 through T6
After T7, the total number of beans remaining in the jars is 5
On T8, it is Sarah’s turn and she must remove 5 beans
After T7, Franco wins if he leaves distributions of
After T7, Franco will lose if he leaves a distribution of
Thus, Franco has a winning strategy if he can ensure that the final 5 beans are not all in one jar, otherwise Sarah has a winning strategy.
Franco has the winning strategy.
We summarize his strategy and then explain why this strategy guarantees
that Franco will always win.
Franco’s strategy:
Remove 1 bean from the jar containing 17 beans on T1
Remove 3 beans from the jar containing the fewest number of beans on T3
Remove beans from the jar containing the greatest number of beans on T5 and T7
On T1, Franco removes 1 bean from the jar containing 17 beans, and so
the distribution is
On T2, Sarah removes 2 beans from one of the jars, and so the
distribution is
On T3, Franco removes 3 beans from the jar containing the fewest number
of beans.
Thus after exactly 3 turns, the jars contain
Through T4 to T7 inclusive,
Thus in each of the above two cases, the jar containing the greatest
number of beans (14 and 16) can not be emptied.
Can Sarah empty a jar that contains 3 beans or that contains 1 bean
without Franco removing any beans from these jars?
Since Sarah can remove 2 or 5 beans on each of her turns, she will be
unable to remove exactly 3 beans and she will similarly be unable to
remove exactly 1 bean.
This means that for each of the above two cases, Sarah will not be able
to empty either of the jars.
Therefore, after exactly 7 turns the jars will contain
There are a total of
As was shown in part (b), 37 beans are removed in each successive group
of 12 turns beginning from the start of the game.
After 109 such groups of turns (a total of
After these 1308 turns, Franco and Sarah are at the beginning of their
sequences and it is Franco’s turn.
With 12 beans remaining, at least one of the jars must still contain 6
or more beans, and so Franco and Sarah were able to remove their
required number of beans on each of the first 1308 turns.
From this point in the game, the winner is determined by how the 12
beans are distributed between the two jars.
For example, assume that the distribution of the final 12 beans is
Franco removes 1, Sarah removes 2, Franco removes 3, Sarah removes 5,
and then there is
On the next turn, Franco is unable to remove 4 beans and so in this case
Sarah wins.
Assume that the distribution of the final 12 beans is
Franco removes 1, and so the jars contain
As we saw in the previous case, if the distribution is
So we assume that Franco removes the 1 bean from the jar containing 11
beans, so that the jars contain
In each of the next 3 turns, beans must be removed from the jar with 10
beans: Sarah removes 2, Franco removes 3, Sarah removes 5, and then
there are
On the next turn, Franco is unable to take 4 beans and so Sarah
wins.
There are other ways that the final 12 beans can be distributed
between the two jars, and for some of these distributions, Sarah is
guaranteed to win.
However, to describe a winning strategy for Sarah concisely, we
demonstrate how Sarah can force the distribution of the final 12 beans
to be one of the two cases above, thus guaranteeing her win.
Next, we demonstrate how Sarah is able to force the distribution of
the final 12 beans to be
Sarah’s winning strategy is to remove all beans from one of the jars, or
to remove all but 1 bean from one of the jars.
Calling this jar the target jar, her strategy is as
follows:
If the target jar contains 5 or more beans, Sarah removes beans from the target jar on each of her turns
If the target jar contains 2, 3 or 4 beans, Sarah removes 2 beans from the target jar (when it’s her turn to remove 2), and removes 5 beans from the other jar (when it’s her turn to remove 5)
Once the target jar contains 0 beans or 1 bean, Sarah removes beans from the other jar on each of her remaining turns
Finally, we explain why Sarah is able to perform this strategy.
In each successive group of 12 turns (6 turns for Sarah), Sarah removes
Thus in the first 109 groups of 12 turns (1308 turns), Sarah removes
Since the jars originally contain 2022 and 2023 beans, Sarah is able to
remove enough beans so that the target jar contains 0 beans or 1
bean.
After the 1308 turns, the other jar contains 12 or 11 beans, which tells
us that Sarah will be able to remove the required number of beans on
each of her turns.
(We note that Franco may also remove beans from the target jar, however
this is okay as it has no effect on Sarah’s strategy.)
Summarizing, Sarah has a strategy that will ensure that the
distribution of the final 12 beans is either
Given that the final 12 beans are distributed in one of these two ways,
Sarah is guaranteed to win this game, and thus Sarah has a winning
strategy.
It is worth noting that the player who has the winning strategy in
this game, Sarah, is unique, however the argument demonstrating her
guaranteed win, is not unique. For example, Sarah is also guaranteed to
win if the distribution of the final 12 beans is