While attempting the latest Internet Puzzle Solvers Test, I found I needed to create a cube. I have graph paper with 1-inch squares, measuring 9 by 7 inches. What is the maximum number of unit cubes I could create from a single page, by cutting out 6-square Cube Nets? Extension: What is the maximum number of unit cubes which could be made from an MxN sheet (M<=N<=10)?
There are 11 different Cube Nets which might be used in building a cube.
I found a listing at the Cube page at MathWorld:
http://mathworld.wolfram.com/Cube.html
Source: Original.
Joseph sent the following configurations:
I found a way to make 9 in a 9x7 sheet. The theoretical maximum of 10 is probably impossible due to the apparently forced 2 wasted squares in two corners. DDEEFFFXX HDDEEEFFF HHDDEIGGX XHXIIIIGX XHHIBAAGG CCCBBBAAG XXCCCBBAA For other sizes, in most cases you can fit N cubes in any rectangle of area 6N+4 or more. 1xN does not fit any cube nets, and likewise for 2x2, 2x3, 2x4, and 3x3. 2xN fits one in 2x5, and one more for each additional three in length, as with 2x8: AAABBBXX XXAAABBB 3xN fits one in 3x4 (several ways), and one more for each additional two in length, as with 3x10: AABBCCDDXX XAABBCCDDX XXAABBCCDD 4x4 fits two cubes: XAAB XABB AABX ABBX 4x6 fits three, and 4x7 fits four by repeating the 4x4 pattern. 4x9 fits five, and 4x10 fits six. 5x5 fits three (without D, below) and 5x6 fits four: ABBBDX AXBCDX AABCDD XABCXD XACCCD The pattern extends by repeating the inner portion to allow 5x8 to fit five cubes, and 5x9 to fit six. 6x7 fits six: XAAXDDD XADDDXE AABEEEE ABBCCEF BBCCFFF BCCFFXX Seven cubes in a 5x10 (a different extension of the other 5xN solutions): ABBBDEEEGX AXBCDXEFGX AABCDDEFGG XABCXDEFXG XACCCDFFFG Seven cubes in 6x8 (an extension of a different four in 6x5 solution, and extendable in 3s by repeating the B-F-G group): AAABBBXX XCAAABBB XCDDEFFG CCDEEFGG CDDEFFGX CDEEFGGX Eight cubes in 6x9: XAACCXEEF XACCEEEFF AABCEGFFH ABBCDGFHH BBDDDGGHX BDDXGGHHX Nine cubes in a 6x10, also extendable in 3s: CCAAABBBXX DCCCAAABBB DHHCIIEFFG DDHHIEEFGG XDHIIEFFGX XDHIEEFGGX Seven in a 7x7: AADDDXX BAAADDD BBAFFXX CBBGFFF CCBGGGF XCEEEGG XCCXEEE Eight in a 7x8: AADDEEXX BAADDEEX BBAADDEE CBBFFGGH CCBFXGHH XCFFGGHX XCCFGHHX Ten in a 7x10 (the rule possibly allows a very tight eleven): AADDEEIIXX BAADDEEIII BBAADDEEHI CBBFGGHHHH CCBFXGGGJH XCFFXXGJJJ XCCFFXJJXX Nine in an 8x8 (the rule possibly allows a very tight ten): AADDEEXX BAADDEEE BBAADDFE CBBGFFFF CBXGGHHF CCIIGGHH XCXIIGHX XCXXIIHX Ten in an 8x9 (eleven may be possible): ADDDEEEXX AAADDDEEE BBAAFFIGG CBFFFIIGH CBBFIIGGH CCBJIXGHH XCJJJXXHX XCXXJJXHX Twelve fit in an 8x10 by doing two 4x10 solutions one above the other. Twelve in a 9x9: ABBJJJXCC AABXJCCCD XABBJCDDD IAABJDDXK IIIIXKKKK IXHHLFEEK HHHGLFFEX HGGGLXFEE GGXLLLFFE Thirteen in a 9x10 (didn't find a fourteen): AAXBBXXXMX DAAABBXXMX DDDACBBLMM EEDDCLLLLM FEEECCKKLM FFFEHCJKKK GXFFHCJJJK GGGHHIIIJJ XXGGHHXIII Fifteen in a 10x10 can be done by placing a nine-in-6x10 solution above a six-in-4x10 solution. The rule allows a very tight sixteen. Probably unsolvable: five in a 5x7, six in a 5x8, five in a 6x6. Unsolved: eleven in a 7x10, ten in an 8x8, eleven in an 8x9, fourteen in a 9x10, sixteen in a 10x10.
I'll assert that you only need the following four: XX XX XX X XXXX X XXX XXX XX XXX X Using these four I've found the following. Up to 6x9 this matches the exhaustive searches. 2 x 10 max 3 best 2 2 x 2 max 0 no solution 2 x 3 max 1 no solution 2 x 4 max 1 no solution 2 x 5 max 1 best 1 optimal 2 x 6 max 2 best 1 2 x 7 max 2 best 1 2 x 8 max 2 best 2 optimal 2 x 9 max 3 best 2 3 x 10 max 5 best 4 3 x 3 max 1 no solution 3 x 4 max 2 best 1 3 x 5 max 2 best 1 3 x 6 max 3 best 2 3 x 7 max 3 best 2 3 x 8 max 4 best 3 3 x 9 max 4 best 3 4 x 10 max 6 best 6 optimal 4 x 4 max 2 best 2 optimal 4 x 5 max 3 best 2 4 x 6 max 4 best 3 4 x 7 max 4 best 4 optimal 4 x 8 max 5 best 4 4 x 9 max 6 best 5 5 x 10 max 8 best 7 5 x 5 max 4 best 3 5 x 6 max 5 best 4 5 x 7 max 5 best 4 5 x 8 max 6 best 5 5 x 9 max 7 best 6 6 x 10 max 10 best 8 6 x 6 max 6 best 4 6 x 7 max 7 best 6 6 x 8 max 8 best 7 6 x 9 max 9 best 7 7 x 10 max 11 best 10 7 x 7 max 8 best 6 7 x 8 max 9 best 8 7 x 9 max 10 best 9 8 x 8 max 10 best 9