MathDB

Initially, there are

3^{80}

particles at the origin

(0,0)

. At each step the particles are moved to points above the

x

-axis as follows: if there are

n

particles at any point

(x,y)

, then

\Bigl \lfloor \frac{n}{3} \Bigr\rfloor

of them are moves to

(x+1,y+1)

\Bigl \lfloor \frac{n}{3} \Bigr\rfloor

are moved to

(x,y+1)

and the remaining to

(x-1,y+1)

, For example, after the first step, there are

3^{79}

particles each at

(1,1),(0,1)

and

(-1,1)

. After the second step, there are

3^{78}

particles each at

(-2,2)

and

(2,2)

2 \times 3^{78}

particles each at

(-1,2)

and

(1,2)

, and

3^{79}

particles at

(0,2)

. After

80

steps, the number of particles at

(79,80)

is:

Problem Statement