tag:blogger.com,1999:blog-5303307482158922565.post631451570912485669..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: Math Teachers at Play #26Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-5303307482158922565.post-24053748023649534982010-05-25T19:50:44.521-07:002010-05-25T19:50:44.521-07:00I held Blaise's comment for a few days, so his...I held Blaise's comment for a few days, so his answer wouldn't spoil people's fun. He and I discussed by email, and he also wrote (and suggested I post):<br /><br />Sloane's Encyclopedia of Integer Sequences describes a slightly<br />different version of the sequence, where it counts the number of<br />points on the surface of a cube where each face is marked with an n+1<br />square lattice of points extending to the edge of the cube (or in my<br />words, the number of distinct visible vertices on an n-Rubik's cube,<br />where a vertex is defined as the (possibly shared) corner of a<br />sub-cube). By this definition, a 0-Cube would have 1 point, a 1-Cube<br />would have 8, etc. Essentially, it shifts the function down by one.<br /><br />This has the pleasing result of changing your n^3-(n-2)^3 to the more<br />symmetric (n+1)^3 - (n-1)^3 and my quadratic formula from 6n^2-12n+8<br />to a more wieldy and easy to remember 6n^2+2 -- both results are<br />listed in the entry<br />http://www.research.att.com/~njas/sequences/A005897Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-37607863725981167792010-05-21T09:30:21.790-07:002010-05-21T09:30:21.790-07:00There are 8 corners, each of which has 1 cube.
Th...There are 8 corners, each of which has 1 cube.<br /><br />There are 12 edges, each of which has n-2 cubes.<br /><br />There are 6 sides, each of which has (n-2)^2 cubes not on edges or corners.<br /><br />So the nth Rubics Cube number is 6(n-2)^2 + 12(n-2) + 8<br /><br />Check: A 2x2x2 would have 6*0^2+12*0+8 = 8, which is correct. A 3x3x3 would have 6*1^2+12*1+8=26. A 4x4x4 would have 6*4+12*2+8 = 24+24+8 = 56. A 5x5x5 would have 6*3^2 + 12*3 + 8 = 54+36+8 = 98.<br /><br />Sounds reasonable to me.<br /><br />Of course, multiplying it all out yields 6n^2 - 12n + 8, I think. Three data points will check it: 2x2x2: 6*4-12*2+8=8, 3x3x3: 6*9-12*3+8=26, 4x4x4: 6*16-12*4+8=96-48+8=56. It's right.Buddha Buckhttps://www.blogger.com/profile/17167036913705912859noreply@blogger.com