bonanova Posted March 12, 2009 Report Share Posted March 12, 2009 Tony Toynbee the toy maker is busy painting the sides of a set of children's building blocks. The blocks are of traditional cubic shape, and Toynbee has paint of five different colors. No two adjacent sides of a block may have the same color. No two blocks in a set may have the same color pattern. How many blocks are there in the largest set Tony can make? Quote Link to comment Share on other sites More sharing options...
0 Gmaster479 Posted March 12, 2009 Report Share Posted March 12, 2009 This reminds me of the whole "How many colors does it take to color the United states so that no 2 borders have the same color" (The answer is 4) This is just that only in 3-D, this might be a while Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 This reminds me of the whole "How many colors does it take to color the United states so that no 2 borders have the same color" (The answer is 4) This is just that only in 3-D, this might be a while My guess would be that each block must have the same color top and bottom - then the sides - 4 option on the first side - three on the second, two on the third - so 5 top / bottoms * 24 options - 120 blocks Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 15 ok, so the top and bottom have to be the same color, leaving us with 4 colors for the remaining sides. Let's label the colors on the sides 1,2,3,and 4. (The top/bottom can be 5). Here is one possible combo: 5 1,2,3,4 5 However, this can be rotated, so it is equivalent to: 5 2,3,4,1 5 It can also be flipped, so another equivalent is: 5 4,3,2,1 5 given this, I only find 3 possible configurations for the sides which are unique: 1,2,3,4 1,2,4,3 1,3,2,4 now, the doubled color (color 5) can be any of the 5 colors. Therefore, we have 5x3 = 15 different blocks Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 12, 2009 Author Report Share Posted March 12, 2009 No one has it yet. Over and under so far. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 well at least we're converging Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 75 Quote Link to comment Share on other sites More sharing options...
0 Prof. Templeton Posted March 12, 2009 Report Share Posted March 12, 2009 bn - Is leaving a face unpainted an option? Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 12, 2009 Author Report Share Posted March 12, 2009 bn - Is leaving a face unpainted an option? No. That would be like a 6th color. A careful read of the OP might help here. I think I have the wording right on this puzzle. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 i think i jumped the gun with my first guess and went way overboard. 25 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 12, 2009 Author Report Share Posted March 12, 2009 i think i jumped the gun with my first guess and went way overboard. 25 Nope. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 (edited) This might be wrong but - 33? Edited March 12, 2009 by scsw Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 Ack! now this is driving me crazy. Is there something wrong with my premise bn? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 I got 210. See pictures for explanation. I didn't draw all the blocks using all five colors at once, so I'll describe it in words: taking one color to be the top and bottom of the block, there are 24 different ways to do the sides. There are 5 colors that can be used for the top and bottom, so 24 * 5 = 120. So the total is 120 + 60 + 30 = 210 I hope I didn't miss any or have any redundancies. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 does each block have to have all 5 colors? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 (edited) I got 210. See pictures for explanation. I didn't draw all the blocks using all five colors at once, so I'll describe it in words: taking one color to be the top and bottom of the block, there are 24 different ways to do the sides. There are 5 colors that can be used for the top and bottom, so 24 * 5 = 120. So the total is 120 + 60 + 30 = 210 I hope I didn't miss any or have any redundancies. Certain ways of folding may yield the same pattern seen from another angle. Edited March 12, 2009 by scsw Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 So i think I have one last guess27 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 I get 60 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 Certain ways of folding may yield the same pattern seen from another angle. I tried to figure out which ones would, and I thought I had eliminated them. But seeing other people's answers, I guess I'm way off. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 Tony Toynbee the toy maker is busy painting the sides of a set of children's building blocks. The blocks are of traditional cubic shape, and Toynbee has paint of five different colors. No two adjacent sides of a block may have the same color. No two blocks in a set may have the same color pattern. How many blocks are there in the largest set Tony can make? 5 blocks largest set possible. Each block has 6 areas( top,bottom,4 sides ) to paint. The first block gets the same color top and bottom and the 4 sides the other 4 colors, the 2nd block a different color top and bottom and the 4 sides the other 4 colors and so on. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 number of blocks = 5 * 4 * 3 * 2 * 1 = 120 ? Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 12, 2009 Author Report Share Posted March 12, 2009 OP: No two blocks in a set may have the same color pattern. This means that two blocks can't be rotated in such a way they are identical. That is, rotating a block doesn't make a new pattern. Be careful not to overcount. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 Tony Toynbee the toy maker is busy painting the sides of a set of children's building blocks. The blocks are of traditional cubic shape, and Toynbee has paint of five different colors. No two adjacent sides of a block may have the same color. No two blocks in a set may have the same color pattern. How many blocks are there in the largest set Tony can make? tony can make infinity boxes if he tries hard enough ☺ Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 Wow. No hits? Tricky one this is. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 new guess: 55 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 12, 2009 Report Share Posted March 12, 2009 51 One way to use three colors. C(5,2)*C(3,2) ways to use four colors = 30 5 * C(4,3) ways to use all five colors = 20 Quote Link to comment Share on other sites More sharing options...
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bonanova
Tony Toynbee the toy maker is busy painting the sides of a set of children's building blocks.
The blocks are of traditional cubic shape, and Toynbee has paint of five different colors.
No two adjacent sides of a block may have the same color.
No two blocks in a set may have the same color pattern.
How many blocks are there in the largest set Tony can make?
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