{"status": "success", "data": {"description_md": "A $3\\times3$ square is partitioned into $9$ unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated $90^\\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?\n\n$\\textbf{(A)}\\ \\dfrac{49}{512}<br>\\qquad\\textbf{(B)}\\ \\dfrac{7}{64}<br>\\qquad\\textbf{(C)}\\ \\dfrac{121}{1024}<br>\\qquad\\textbf{(D)}\\ \\dfrac{81}{512}<br>\\qquad\\textbf{(E)}\\ \\dfrac{9}{32}$\n___\nFull 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\">3\\times3</span>  square is partitioned into  <span class=\"katex--inline\">9</span>  unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated  <span class=\"katex--inline\">90^\\circ</span>  clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\dfrac{49}{512}\\qquad\\textbf{(B)}\\ \\dfrac{7}{64}\\qquad\\textbf{(C)}\\ \\dfrac{121}{1024}\\qquad\\textbf{(D)}\\ \\dfrac{81}{512}\\qquad\\textbf{(E)}\\ \\dfrac{9}{32}</span> </p>&#10;<hr><p>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": 3, "problem_name": "2012 AMC 12A Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p16", "prev": "/problem/12_amc12A_p14"}}