Problem 1478

Solution:

We begin by filling the $10 \times 10$ grid formed by rows and columns $1$ through $10$ with any values, which we can do in $\left(2^{10}\right)^{100} = 2^{1000}$ ways. Then in column $0$, there is at most $1$ way to fill in the square in row $10$, $2$ ways for the square in row $9$, down to $2^{10}$ ways in row $0$. Similarly, there is $1$ way to fill in the square in row $0$ and column $10$, $2$ ways to fill in the square in row $0$ and column $9$, etc. Overall, the number of ways to fill out the squares in row or column $0$ is $2^{1} \cdot 2^{2} \cdot 2^{3} \cdots 2^{9} \cdot 2^{10} \cdot 2^{9} \cdot 2^{8} \cdots 2^{1} = 2^{100}$, so the number of possible distinct grids is $2^{1000} \cdot 2^{100} = 2^{1100}$.