2020 CIME I Problems/Problem 1

Problem 1

A knight begins on the point $(0,0)$ in the coordinate plane. From any point $(x,y)$ the knight moves to either $(x+2,y+1)$ or $(x+1,y+2)$. Find the number of ways the knight can reach the point $(15,15)$.


Let $A$ denote a move of $2$ units north and $1$ unit east, and let $B$ denote a move of $1$ unit north and $2$ units east. To get to the point $(15,15)$ using only these moves, say $a$ moves in direction $A$ and $b$ moves in direction $B$, we must have $2a+1b=1a+2b=15$ because both the $x$- and $y$-coordinates have increased by $15$ since the knight started. Solving this system of equations gives us $a=b=5$. This means we need the knight to make $10$ moves, $5$ of which are headed in direction $A$, and the remaining $5$ are headed in direction $B$. Any combination of these moves work, so the answer is $\binom{10}{5}=\boxed{252}.$

Video Solution

https://www.youtube.com/watch?v=SFVt0JYLkHY ~Shreyas S

See also

2020 CIME I (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. AMC logo.png