bonanova Posted February 6, 2009 Report Share Posted February 6, 2009 Take a cube with 5" edges, and paint it blue. Then slice it mathematically [zero saw blade thickness] into 125 unit cubes. Throw away any cube that has no blue faces, and construct a rectangular brick with an all-blue exterior. How large [volume] can that brick be? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 Does the brick have to be solid? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 Sorry if this is a bit nonsensical (it's late and I'm tired), but 96 cubic inches, a prism of dimensions 4x4x6. The interior of a 5x5x5 cube is 3x3x3, or 27 cubic inches, so the maximum volume of a shape built from the remaining blocks is 98 cubic inches. However, a prism of that volume must be of the dimensions 2x7x7, and thus must contain 48 blocks with two or more sides colored blue, whereas the original 5x5x5 only gives us 44 such blocks to work with. Therefore, 96 is the maximum possible volume, and in the case of a 4x4x6 prism, the solution is achieved. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 (edited) Let's see... OK, I get 98 cubes with "blue" on them - Each face has 25 pieces. We have 8 corner pieces with paint on three sides (we will need all 8 of those), 12 edges with three cubes each with 2 sides of paint = 36, which leaves 54 "single sided" paint cubes. A rectangle of any size needs the eight corner cubes to be "all painted." Sides of 6 = 4 each of the double-sided ones = 16 (leaves 20 of the doubles) Sides of 7 = 5 each of the double sided ones = 20 (leaves none of the doubles) Sides of 2 = no doubles needed there... 40 inside ones (4*5*2) Would make 84... Is there anything bigger? Edited February 6, 2009 by IdoJava Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 Sorry if this is a bit nonsensical (it's late and I'm tired), but 96 cubic inches, a prism of dimensions 4x4x6. The interior of a 5x5x5 cube is 3x3x3, or 27 cubic inches, so the maximum volume of a shape built from the remaining blocks is 98 cubic inches. However, a prism of that volume must be of the dimensions 2x7x7, and thus must contain 48 blocks with two or more sides colored blue, whereas the original 5x5x5 only gives us 44 such blocks to work with. Therefore, 96 is the maximum possible volume, and in the case of a 4x4x6 prism, the solution is achieved. I have to agree with hookemhorns' answer Out of the 125 cubes, only 98 will have blue on at least one side. 8 will have blue on 3 sides, 36 will have blue on 2 sides and 54 have blue on only 1 side. So, 98 is the max Vol. and the only way to make that prism is 7x7x2. This will reqire 8 - 3 siders, 40 - 2 siders and 50 1-siders. We don't have 40 - 2 siders. 97 is prime, so no dice. 96 can be made with a multitude of dimensions, but 4x4x6 will reduce the length of the edges. This prism will require 8 - 3 siders, 32 - 2 siders, 40 - 1 siders and 16 misc. cubes. This is possible and the max Vol. for the problem. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 25 pieces on the top and bottom 15 pieces of 2 sides 9 remaining faces on other two sides gives you 98 sides painted blue of these 98 side there are 8 pieces are corners 24 pieces edges 54 pieces have 1 colored sides volume = 86 cc Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 Take a cube with 5" edges, and paint it blue. Then slice it mathematically [zero saw blade thickness] into 125 unit cubes. Throw away any cube that has no blue faces, and construct a rectangular brick with an all-blue exterior. How large [volume] can that brick be? I guess I'll do like the OP says and toss the 3x3 internal, blue-less cubes. That leaves me the original shell of the 5x5x5 (125" cube) brick. I'll just stack those up, bottom and sides are easy. But how to support the roof of the hollow structure? Well, I'll just use those zero-width saw blades! My cube will be 125" cube on the outside, with a 27" cube interior volume. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted February 6, 2009 Author Report Share Posted February 6, 2009 Does the brick have to be solid? Yes. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted February 6, 2009 Author Report Share Posted February 6, 2009 I guess I'll do like the OP says and toss the 3x3 internal, blue-less cubes. That leaves me the original shell of the 5x5x5 (125" cube) brick. I'll just stack those up, bottom and sides are easy. But how to support the roof of the hollow structure? Well, I'll just use those zero-width saw blades! My cube will be 125" cube on the outside, with a 27" cube interior volume. Cute, but don't build yer house using them... <_< Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 I have to agree with hookemhorns' answer Out of the 125 cubes, only 98 will have blue on at least one side. 8 will have blue on 3 sides, 36 will have blue on 2 sides and 54 have blue on only 1 side. So, 98 is the max Vol. and the only way to make that prism is 7x7x2. This will reqire 8 - 3 siders, 40 - 2 siders and 50 1-siders. We don't have 40 - 2 siders. 97 is prime, so no dice. 96 can be made with a multitude of dimensions, but 4x4x6 will reduce the length of the edges. This prism will require 8 - 3 siders, 32 - 2 siders, 40 - 1 siders and 16 misc. cubes. This is possible and the max Vol. for the problem. These are the best answer, You take your saw with the zero width blade, cut 14 of the blocks in half (including 2 of the 2-color ones). You can then construct a 3.5 x 4 x 7 cube with a blue exterior and 98 cu. in. volume. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 6, 2009 Report Share Posted February 6, 2009 funny..looking at this makes me hungry..and dizzy Quote Link to comment Share on other sites More sharing options...
0 Guest Posted February 8, 2009 Report Share Posted February 8, 2009 These are the best answer, You take your saw with the zero width blade, cut 14 of the blocks in half (including 2 of the 2-color ones). You can then construct a 3.5 x 4 x 7 cube with a blue exterior and 98 cu. in. volume. I agree with Xucam. It is never said that we need whole numbered dimensions, it can also be a fraction. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted February 12, 2009 Author Report Share Posted February 12, 2009 I agree with Xucam. It is never said that we need whole numbered dimensions, it can also be a fraction. The OP says saw, discard, and construct, in that order. But you have a point that it does not specifically prohibit further sawing. This is the biggest size possible if more sawing is permitted.... Quote Link to comment Share on other sites More sharing options...
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bonanova
Take a cube with 5" edges, and paint it blue. Then slice it
mathematically [zero saw blade thickness] into 125 unit cubes.
Throw away any cube that has no blue faces, and
construct a rectangular brick with an all-blue exterior.
How large [volume] can that brick be?
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