tag:blogger.com,1999:blog-5730391.post5106492001936754068..comments2024-04-12T03:59:51.417-07:00Comments on Blaine's Puzzle Blog: Playing with BlocksBlainehttp://www.blogger.com/profile/06379274325110866036noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5730391.post-61804819164194478292009-11-24T23:50:29.436-08:002009-11-24T23:50:29.436-08:00Another approach: in order to attain 45 unstained...Another approach: in order to attain 45 unstained blocks, the obvious dimension of the region of their source is a 3 x 3 x 5 block segment. since the min dimension which would have a 3 x 3 interior of a stained perimeter would be 5 x 5, and the height of the proposed segment would be 5, the initial block must be 5 x 5 x 5 and in order to be 5 high, both ends must be unstained.Unknownhttps://www.blogger.com/profile/09703729276485900064noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-52000345364330406892008-04-18T10:35:00.000-07:002008-04-18T10:35:00.000-07:00The number of cubes on a side has to be between 4 ...The number of cubes on a side has to be between 4 and 5. Any less and you don't have enough smaller cubes. Any more and you have too many unpainted cubes in the middle.<BR/><BR/>You can try to use a 4x4x4 (which has 8 unpainted cubes on the interior), but you won't find a way to paint some of the sides to leave 37 fully unpainted cubes.<BR/><BR/>With a 5x5x5 cube, you have 27 in the interior and you need 18 more. The way to do that is to leave two ends unpainted. That will result in 9 more on each face that are fully unpainted.<BR/><BR/>Answer:<BR/>5x5x5 cube with two opposite faces unpainted. 80 cubes with paint, 45 without.Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-31312126917695085882008-03-24T23:31:00.000-07:002008-03-24T23:31:00.000-07:00Anyone else working on this? Or should I just pos...Anyone else working on this? Or should I just post the answer?Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-58518179600107973342008-03-16T22:20:00.000-07:002008-03-16T22:20:00.000-07:00I got it, but I won't post the answer just yet. I...I got it, but I won't post the answer just yet. <BR/><BR/>If you make "m" cuts in each direction, then each face will be divided a grid (m+1) by (m+1).<BR/><BR/>There will be (m+1)^3 small cubes, of which (m-1)^3 will be internal cubes and must be unpainted.<BR/><BR/>Based on this information, there are only 2 possible values for "m" such that 45 cubes remain unpainted.EricMargelhttps://www.blogger.com/profile/02090556952649043090noreply@blogger.com