Irska

Following the hint, we observe that the ball lands in a pocket as soon as the line segment reaches a point $(c,c)$ where $c$ is divisible by both $a$ and $b$; clearly $c$ is the least common multiple of $a$ and $b$. The ball touches a vertical rail $c$ times and a horizontal rail $c$ times. If $c$ is even/odd, the ball lands in a pocket on the left/right side of the table, and if $c$ is even/odd, the ball lands in a pocket on the lower/upper side of the table. Hence the ball lands in the upper left pocket exactly when $c$ is even and $c$ is odd.



Write $a=2^j{a}^{\prime }$ and $b=2^k{b}^{\prime }$, where $a^{\prime }$ and $b^{\prime }$ are odd. Then $c=2^{\max (j,k)}\mathrm{lcm}(a^{\prime },b^{\prime })$. Since $\mathrm{lcm}(a^{\prime },b^{\prime })$ is odd, $c$ is even and $c$ is odd precisely when $j<k$.

If $a$ is odd, then the ball lands in the upper left pocket whenever $b$ is even; there are $\frac{a-1}{2}$ such allowed values of $b$. If $a$ is divisible by $2$ but not by $4$, then the ball lands in the upper left pocket whenever $b$ is divisible by $4$; there are $\frac{a-2}{4}$ such allowed values of $b$, and so forth.

Fix an integer $N$ and count the ordered pairs $(a,b)$ with $1\le b\le a\le N$ for which the ball lands in the upper left pocket. In the calculation we use $C(k,2)=\left(\genfrac{}{}{0ex}{}{k}{2}\right)=0+1+2+\cdots +(k-2)+(k-1)$.

When $a=1,3,5,7,\dots$, then there are $0,1,2,3,\dots$ allowed values of $b$; summing over all such $a$ gives us $C\left(\left[\frac{N+1}{2}\right],2\right)$ total ordered pairs. When $a=2,6,10,14,\dots$, there are $0,1,2,3,\dots$ allowed values of $b$; summing over all such $a$ gives us $C\left(\left[\frac{N+2}{4}\right],2\right)$ total ordered pairs, and so forth. For $N=2009$, there are a total of $C(1005,2)+C(502,2)+C(251,2)+C(126,2)+C(63,2)+C(31,2)+C(16,2)+C(8,2)+C(4,2)+C(2,2)=672,084$ allowed ordered pairs, so this is our answer.