{"status": "success", "data": {"description_md": "A $7 \\times 1$ board is completely covered by $m \\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \\times 1$ board in which all three colors are used at least once. For example, a $1 \\times 1$ red tile followed by a $2 \\times 1$ green tile, a $1 \\times 1$ green tile, a $2 \\times 1$ blue tile, and a $1 \\times 1$ green tile is a valid tiling. Note that if the $2 \\times 1$ blue tile is replaced by two $1 \\times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A <span class=\"katex--inline\">7 \\times 1</span> board is completely covered by <span class=\"katex--inline\">m \\times 1</span> tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let <span class=\"katex--inline\">N</span> be the number of tilings of the <span class=\"katex--inline\">7 \\times 1</span> board in which all three colors are used at least once. For example, a <span class=\"katex--inline\">1 \\times 1</span> red tile followed by a <span class=\"katex--inline\">2 \\times 1</span> green tile, a <span class=\"katex--inline\">1 \\times 1</span> green tile, a <span class=\"katex--inline\">2 \\times 1</span> blue tile, and a <span class=\"katex--inline\">1 \\times 1</span> green tile is a valid tiling. Note that if the <span class=\"katex--inline\">2 \\times 1</span> blue tile is replaced by two <span class=\"katex--inline\">1 \\times 1</span> blue tiles, this results in a different tiling. Find the remainder when <span class=\"katex--inline\">N</span> is divided by <span class=\"katex--inline\">1000</span>.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2013 AIME II Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/13_aime_II_p10", "prev": "/problem/13_aime_II_p08"}}