LatticeMethod Exeter Difficult
2014
Problem - 3328
While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first
point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For
example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later
in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically
reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a
player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his
or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and
finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How
many possible games could they have played? Two games are considered the same if and only if they
include the exact same sequence of scoring.
Answer
120
This is a challenging problem, but can be solved using the lattice method using a $22\times 22$ grid illustrated below. The $X$-axis shows Chad's score while the $Y$-axis shows Jordan's. We then place a stop/blockage on all the grid points where the pair of scores can be simplified.
Then each path shows a possible progress of a game. When a path reaches the right boundary, Chad wins the game. Otherwise, when a path ends at any point on the upper boundary, Jordan wins the game. The whole completed grid will look like the below:
Adding all the numbers on the right and upper boundary gives us the answer as $$(2+14+22+22)\times 2 = \boxed{120}$$