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?

Question

Homi · Accepted Answer

Adding up hexagonal numbers gives a sequence of cubic numbers.

Step 1 — Calculate the sums

Let us add the given hexagonal numbers one by one.

The first hexagonal number is 1. $\text{Sum of first hexagonal number} = 1$

$\boxed{1}$

Next, we add the first two hexagonal numbers. $\text{Sum of first two hexagonal numbers} = 1 + 7$ $= 8$

$\boxed{8}$

Then, we add the first three hexagonal numbers. $\text{Sum of first three hexagonal numbers} = 1 + 7 + 19$ $= 27$

$\boxed{27}$

Finally, we add the first four hexagonal numbers. $\text{Sum of first four hexagonal numbers} = 1 + 7 + 19 + 37$ $= 64$

$\boxed{64}$

Diagram 1

Step 2 — Identify the sequence

The sums we found are 1, 8, 27, 64. This sequence is $1 \times 1 \times 1$ , $2 \times 2 \times 2$ , $3 \times 3 \times 3$ , $4 \times 4 \times 4$ . These are cubic numbers ( $1^3, 2^3, 3^3, 4^3$ ).

Step 3 — Explain with the cube picture

Look at the cube picture. It shows a large cube. This large cube is made of many small unit cubes. The cube in the picture is a $4 \times 4 \times 4$ cube. It has $4 \times 4 \times 4 = \textbf{64}$ small cubes in total.

We can think of building this large cube layer by layer. First, we start with a tiny $1 \times 1 \times 1$ cube. This has 1 small cube. This is the first hexagonal number.

Next, we add cubes around the $1 \times 1 \times 1$ cube. We make a $2 \times 2 \times 2$ cube. We added $2^3 - 1^3 = 8 - 1 = \textbf{7}$ new cubes. This is the second hexagonal number.

Then, we add more cubes around the $2 \times 2 \times 2$ cube. We make a $3 \times 3 \times 3$ cube. We added $3^3 - 2^3 = 27 - 8 = \textbf{19}$ new cubes. This is the third hexagonal number.

Finally, we add even more cubes around the $3 \times 3 \times 3$ cube. We make a $4 \times 4 \times 4$ cube. We added $4^3 - 3^3 = 64 - 27 = \textbf{37}$ new cubes. This is the fourth hexagonal number.

The total number of cubes in the $4 \times 4 \times 4$ cube is 64. This total is exactly the sum of the hexagonal numbers we added: $1 + 7 + 19 + 37 = \textbf{64}$ . Each hexagonal number represents a "shell" of cubes. When we add these shells, they build up a complete cube.

Diagram 2

Answer

When you start to add up hexagonal numbers, you obtain a sequence of cubic numbers.

(i) The sums are 1, 8, 27, 64. (ii) This sequence is the cubic numbers ( $1^3, 2^3, 3^3, 4^3$ ). (iii) Each hexagonal number represents a layer of cubes. Adding these layers builds a larger cube. For example, $1+7+19+37 = 64$ , which is a $4 \times 4 \times 4$ cube.

Question 17

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