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Mathematical Olympiad in Ukraine, Third Round (January 23, ), 2010

Geometry Difficulty 6.0 National Olympiad Prove it Ukraine

Cube with side length 1111 is partitioned into unit cubes. The external slice of cubes is colored with yellow, the next slice (all cubes that share a vertex with the external slice) is colored with blue. Slice of cubes that touches the previous slice is colored with yellow and so on. Find the number of yellow and blue cubes.

Solution

Obviously, there are 66 slices of cubes. It is easy to see that the external yellow slice has height 1111, the next blue - 99, the next yellow - 77, the next blue - 55, the next yellow - 33 and the last one (6th slice) blue has height 11, e.g. it's a unit cube. Further, we can calculate the number of cubes in each slice in such a way:

5th slice, yellow, with height 33, has 33−13=263^3 - 1^3 = 26 unit cubes.

4th slice, blue, with height 55, has 53−33=985^3 - 3^3 = 98 unit cubes.

3rd slice, yellow, with height 77, has 73−53=2187^3 - 5^3 = 218 unit cubes.

2nd slice, blue, with height 99, has 93−73=3869^3 - 7^3 = 386 unit cubes.

1st slice, yellow, with height 1111, has 113−93=60211^3 - 9^3 = 602 unit cubes.

Finally, the number of blue cubes is 1+98+386=4851 + 98 + 386 = 485, and the number of yellow cubes is 602+218+26=846602 + 218 + 26 = 846.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.