Odd Sum Rectangles

AtCoder

IDhitachi2020_e

Time2000ms

Memory256MB

Difficulty—

We have a grid with $(2^N - 1)$ rows and $(2^M-1)$ columns. You are asked to write $0$ or $1$ in each of these squares. Let $a_{i,j}$ be the number written in the square at the $i$\-th row from the top and the $j$\-th column from the left. For a quadruple of integers $(i_1, i_2, j_1, j_2)$ such that $1\leq i_1 \leq i_2\leq 2^N-1, 1\leq j_1 \leq j_2\leq 2^M-1$, let $S(i_1, i_2, j_1, j_2) = \displaystyle \sum_{r=i_1}^{i_2}\sum_{c=j_1}^{j_2}a_{r,c}$. Then, let the _oddness_ of the grid be the number of quadruples $(i_1, i_2, j_1, j_2)$ such that $S(i_1, i_2, j_1, j_2)$ is odd. Find a way to fill in the grid that maximizes its oddness. ## Constraints * $N$ and $M$ are integers between $1$ and $10$ (inclusive). ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

1 2

Output #1

111

For this grid, $S(1, 1, 1, 1)$, $S(1, 1, 2, 2)$, $S(1, 1, 3, 3)$, and $S(1, 1, 1, 3)$ are odd, so it has the oddness of $4$.
We cannot make the oddness $5$ or higher, so this is one of the ways that maximize the oddness.

API Response (JSON)

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      "content": "We have a grid with $(2^N - 1)$ rows and $(2^M-1)$ columns. You are asked to write $0$ or $1$ in each of these squares. Let $a_{i,j}$ be the number written in the square at the $i$\\-th row from the to...",
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