{"status": "success", "data": {"description_md": "A $3 \\times 3$ 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 the grid is now entirely black?\n\n$\\textbf{(A)}\\ \\frac{49}{512}\\qquad\\textbf{(B)}\\ \\frac{7}{64}\\qquad\\textbf{(C)}\\ \\frac{121}{1024}\\qquad\\textbf{(D)}\\ \\frac{81}{512}\\qquad\\textbf{(E)}\\ \\frac{9}{32}$", "description_html": "<p>A  <span class=\"katex--inline\">3 \\times 3</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 the grid is now entirely black?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{49}{512}\\qquad\\textbf{(B)}\\ \\frac{7}{64}\\qquad\\textbf{(C)}\\ \\frac{121}{1024}\\qquad\\textbf{(D)}\\ \\frac{81}{512}\\qquad\\textbf{(E)}\\ \\frac{9}{32}</span> </p>\n<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 10A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc10A_p21", "prev": "/problem/12_amc10A_p19"}}