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 $(15,15)$ .

Solution

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 $x$ moves in direction $A$ and $y$ moves in direction $B$ , we must have $2x+1y=1x+2y=15$ because both the $x$ and $y$ -coordinates have increased by $15$ since the knight started. Solving this system of equations gives us $x=y=5$ . 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 $\binom{10}{5}=\boxed{252}.$

2020 CIME I (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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2020 CIME I Problems/Problem 1

Problem 1

Solution