2010 AMC 12A Problem 18

Question

A

16

-step path is to go from

(-4,-4)

to

(4,4)

with each step increasing either the

x

-coordinate or the

y

-coordinate by

1.

How many such paths stay outside or on the boundary of the square

-2\le x\le2,

-2\le y\le2

at each step?

92

144

1568

1698

12{,}800

Each step increases

x+y

by

1,

so the path meets the line

x+y=0

at exactly one lattice point

To avoid the square’s interior, that point is

(\pm2,\mp2),(\pm3,\mp3),

or

(\pm4,\mp4);

count paths through each

Every step increases

x+y

by

1,

which runs from

-8

to

8,

so each path passes through exactly one lattice point with

x+y=0.

To stay out of the open square, that point

(t,-t)

must have

|t|\ge2,

so it is one of

(\pm2,\mp2),(\pm3,\mp3),(\pm4,\mp4).

By symmetry consider the three points

(-4,4),(-3,3),(-2,2)

and double. The number of paths from

(-4,-4)

to

(-(4-j),4-j)

is

\binom{8}{j},

and the number continuing on to

(4,4)

is also

\binom{8}{j}.

Therefore the total is

\begin{gathered} 2\left(\binom80^2+\binom81^2+\binom82^2\right) \\ =2(1+64+784) \\ =1698. \end{gathered}

Thus, D is the correct answer.

Problem 18 in other years