# LR Visualisation: Cube Puzzle

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## LR Visualisation Puzzle: Cube

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.

On the bottom-most square layer of the cube, consider the surface diagonal that has the square numbered 8, find the sum of all the numbers on the cubes on this surface diagonal on the bottom-most layer.

260

## Video Solution

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## Detailed Solution

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 1
Since this diagonal is on the bottom-most layer, we can refer to the diagram above. Clearly the diagonal is made up of numbers forming an arithmetic progression, where the first term is 8, and the common difference is 7. There are 8 terms in all. We can find the sum of all the numbers using the AP formula:

${Sum = (2a + (n-1)d) \frac{n}{2} }$

${Sum = (2 \times 8+7 \times 7) \times 4=260 }$

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