What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ...? Which sequence do you get? Can you explain it using a picture of a cube?
When you add up hexagonal numbers, you get:
1 = 1 1 + 7 = 8 1 + 7 + 19 = 27 1 + 7 + 19 + 37 = 64 1 + 7 + 19 + 37 + 61 = 125
This gives the sequence: 1, 8, 27, 64, 125, ...
These are the Cube numbers sequence (1³, 2³, 3³, 4³, 5³, ...).
Pictorial explanation using cubes:
The first hexagonal number 1 represents one unit cube (1³ = 1).
The sum of first two hexagonal numbers 1 + 7 = 8 represents a ×2 cube made of 8 unit cubes.
The sum of first three hexagonal numbers 1 + 7 + 19 = 27 represents a ×3 cube made of 27 unit cubes.
Each successive cube can be built by adding a new layer of hexagonal number of cubes around the previous cube.
Explanation
The hexagonal numbers are 1, 7, 19, 37, 61, 91, ... Their cumulative sums are 1, 8, 27, 64, 125, 216, ... which are perfect cubes (1³, 2³, 3³, 4³, 5³, 6³, ...). This remarkable relationship shows that hexagonal numbers are the differences between consecutive cubes. The pictorial explanation involves building cubes layer by layer, where each layer added corresponds to a hexagonal number.
Solution Steps
Calculate cumulative sums of hexagonal numbers: 1, 1+7=8, 1+7+19=27, 1+7+19+37=64, 1+7+19+37+61=125
Identify the resulting sequence: 1, 8, 27, 64, 125, ... which are perfect cubes (1³, 2³, 3³, 4³, 5³, ...)
Explain pictorially: Each cumulative sum represents a cube built by adding layers of hexagonal numbers around the previous cube