Rubo Cubo

How many unit cubes are visible on a 7 × 7 × 7 7\times 7\times 7 cube?

For example,

  • the 2 × 2 × 2 2\times 2\times 2 cube has 8 visible unit cubes;
  • the 3 × 3 × 3 3\times 3\times 3 cube has 26 visible unit cubes;
  • the 4 × 4 × 4 4\times 4\times 4 cube has 56 visible unit cubes;


The answer is 218.

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3 solutions

Siva Budaraju
Mar 25, 2017

For any cube with dimensions n n n and n>1, there will be a (n-2) (n-2) (n-2) cube inside it and just barely not reaching. The number of cubes visible is the area of the big cube - the area of the contained cube. In this case, it will be 7^3-5^3 = 218 cubes visible.

Marta Reece
Mar 22, 2017

A cube has 6 6 faces, 12 12 edges, and 8 8 vertices.

A face has 7 × 7 = 49 7\times 7=49 cubes, so that will be total of 6 × 49 = 294 6\times49=294 .

But the edges have been counted with both sides, so they need to be subtracted.

An edge has 7 cubes, so that is a total of 7 × 12 = 84. 7\times12=84. Subtract that and we have 294 84 = 210 294-84=210 .

But what about the vertices? They were each counted three times in the first round, and each was subtracted three times in the second, so they need to be counted. There are 8 8 of them, each with only 1 1 cube. So the final correction is 210 + 8 = 218 210+8=218 .

Dilwar Ali Alvee
Mar 16, 2017

According to the question, 2x2x2 = 8 pieces 3x3x3 = 26 pieces 4x4x4= 56 pieces If "n" is the amount of piece of a side of a cube (like n=3 for 3x3x3) Then, n=2, 2x2x2 = 8 = 2^3 - (n-2)^3 = 2^3 - (2-2)^3 n=3, 3x3x3 = 26 = 3^3 - (n-2)^3 = 3^3 - (3-2)^3 n=4, 4x4x4 = 56 = 4^3 - (n-2)^3 = 4^3 - (4-2)^3 So, for 7x7x7, n=7 7x7x7 = 7^3 - (n-2)^3 =7^3 - (7-2)^3 =7^3 - 5^3 =343 - 125 =218

Ans:218 pieces.

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