A large cube, of total volume 512 cm3 is made up of smaller 1 cm3 cubes. The larger cube is made by following these rules:
1. Start from the left hand side, and number the small cubes 1 to 8, from left to right.
2. Place cube no. 9 behind cube no. 1 to start the second row, and proceed all the way to cube no. 64
3. Start the second layer on top of cube no 1, and build the second layer from left to right, and front to back like the first layer.
General Solution
The first layer of the cube looks like this:
We notice that each of the cubes in the second layer will be 64 + n, where n is the small cube directly below the cube in question. Each of the cubes in the third layer will be 64 x 2 + n, and so on.
In general, to find the number on a particular cube 'P' on the k'th layer, we can use the formula P = (k - 1)x64 + n, where n is the number on the bottom-most cube below P. With this general idea, we can solve the whole puzzle.
Solution for Question 4
The body diagonal that has the cube numbered 1 starts in the bottom-left-front corner and extends all the way to the top-right-back corner. Let us first look at the diagonal that extends from cube numbered 1 to cube numbered 64. We can use the diagram for this. On the body diagonal, the first cube will be 1, the second cube will be 10 + 64, the third cube will be 19 + 64 +64 and so on.
The sum will be:
Correct Answer: 2052.
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