What cubes end in, and what they add up to

Odd numbers added in order from 1 made squares. In groups, they make cubes too.

Rule: Take the odd numbers in order. Group them 1, then 2, then 3, and so on. The n-th group adds up to n cubed.

Here n is the number of the group. The 3rd group is 7 + 9 + 11 = 27 = 3³.

The first ten groups use 55 odd numbers. The 55th odd number is 109, so group 11 starts at 111. The eleven odd numbers from 111 to 131 form the eleventh group. Their sum is 11³ = 1331.

Every digit appears once as a last digit. So no last digit rules out a cube. Zeros at the end of a cube come in threes. 1000 has three. 100 has two, so it is not a cube.

Slide n. Add the odd numbers. See the cube.