Problem 126

Cuboid layers

The minimum number of cubes to cover every visible face on a cuboid measuring 3 x 2 x 1 is twenty-two.

If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred and eighteen cubes to cover every visible face.

However, the first layer on a cuboid measuring 5 x 1 x 1 also requires twenty-two cubes; similarly the first layer on cuboids measuring 5 x 3 x 1, 7 x 2 x 1, and 11 x 1 x 1 all contain forty-six cubes.

We shall define C(n) to represent the number of cuboids that contain n cubes in one of its layers. So C(22) = 2, C(46) = 4, C(78) = 5, and C(118) = 8.

It turns out that 154 is the least value of n for which C(n) = 10.

Find the least value of n for which C(n) = 1000.

立方体层

要完全包住一个3x2x1的长方体的表面，最少需要的立方体数目为22。

如果我们要再包一层，需要46个立方体才能挡住所有表面，而第三层需要78个立方体，第4层则需要118个立方体。

同样地，要完全包住一个5x1x1的长方体的表面也需要22个立方体。而要包住5x3x1或是7x2x1或是11x1x1的长方体表面，第一层就需要46个立方体。

记C(n)是在任意一层中需要n个立方体的长方体数目。所以C(22) = 2，C(46) = 4，C(78) = 5，C(118) = 8。

可以验证154是第一个使得C(n) = 10的n。

找出使得C(n) = 1000的最小的n。