# 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 big cube, there exists an external surface whose four corner cubes have the numbers 8, 64, 449 and 512. Find the sum of all the numbers on the cubes on this surface.

16640

## 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 5
This is a super hard question, because it involves a lot of calculation. Basically, we have to add all columns of cubes on the right-most layer of the larger cube. Each column is an AP with first term the number on the bottom layer, and common difference 64.

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

Sum for First Column:${Sum = (2 \times 8 + (8-1)64) \times 4=1856 }$

Sum for Second Column:${Sum = (2 \times 16 + (8-1)64) \times 4=1920 }:$

Sum for Third Column:${Sum = (2 \times 24 + (8-1)64) \times 4=1984 }:$

We can quickly see the pattern (that each column exceeds the previous by 64).
Sum of all columns: ${Sum = (2 \times 1856 + (8-1)64) \times 4=16640 }$ Correct Answer: 16640.

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