MathDB

We have twenty-seven

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cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one

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cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of all the

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cubes. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,7)--(-1,4)); draw((-1,9.15)--(-3.42,8.21)); draw((-1,9.15)--(1.42,8.21)); draw((-1,7)--(1.42,8.21)); draw((1.42,7.21)--(-1,6)); draw((1.42,6.21)--(-1,5)); draw((1.42,5.21)--(-1,4)); draw((1.42,8.21)--(1.42,5.21)); draw((-3.42,8.21)--(-3.42,5.21)); draw((-3.42,7.21)--(-1,6)); draw((-3.42,8.21)--(-1,7)); draw((-1,4)--(-3.42,5.21)); draw((-3.42,6.21)--(-1,5)); draw((-2.61,7.8)--(-2.61,4.8)); draw((-1.8,4.4)--(-1.8,7.4)); draw((-0.2,7.4)--(-0.2,4.4)); draw((0.61,4.8)--(0.61,7.8)); label("2",(-1.07,9.01),SE*labelscalefactor); label("9",(-1.88,8.65),SE*labelscalefactor); label("1",(-2.68,8.33),SE*labelscalefactor); label("3",(-0.38,8.72),SE*labelscalefactor); draw((-1.8,7.4)--(0.63,8.52)); draw((-0.27,8.87)--(-2.61,7.8)); draw((-2.65,8.51)--(-0.2,7.4)); draw((-1.77,8.85)--(0.61,7.8)); label("7",(-1.12,8.33),SE*labelscalefactor); label("5",(-1.9,7.91),SE*labelscalefactor); label("1",(0.58,8.33),SE*labelscalefactor); label("18",(-0.36,7.89),SE*labelscalefactor); label("1",(-1.07,7.55),SE*labelscalefactor); label("1",(-0.66,6.89),SE*labelscalefactor); label("5",(-0.68,5.8),SE*labelscalefactor); label("1",(-0.68,4.83),SE*labelscalefactor); label("2",(0.09,7.27),SE*labelscalefactor); label("1",(0.15,6.24),SE*labelscalefactor); label("2",(0.11,5.26),SE*labelscalefactor); label("1",(0.89,7.61),SE*labelscalefactor); label("3",(0.89,6.63),SE*labelscalefactor); label("9",(0.92,5.62),SE*labelscalefactor); label("18",(-3.18,7.63),SE*labelscalefactor); label("2",(-3.07,6.61),SE*labelscalefactor); label("2",(-3.09,5.62),SE*labelscalefactor); label("1",(-2.29,7.25),SE*labelscalefactor); label("3",(-2.27,6.22),SE*labelscalefactor); label("5",(-2.29,5.2),SE*labelscalefactor); label("7",(-1.49,6.89),SE*labelscalefactor); label("34",(-1.52,5.81),SE*labelscalefactor); label("1",(-1.41,4.86),SE*labelscalefactor); [/asy]

Problem Statement