Cube numbers and their patterns

If a square is what you see when a number multiplies itself once, a cube is what you see when it multiplies itself twice. Picture a die: each side has the same length, and the total number of tiny unit-cubes inside is the cube of that length. The list $1, 8, 27, 64, 125, 216, \dots$ may look ordinary but it sits at the heart of volume, growth, and one of the most famous anecdotes in modern mathematics.

Concept

A whole number $n$ is a perfect cube if $n = k \times k \times k = k^3$ for some whole number $k$. So $64 = 4^3$ is a cube, but $50$ is not.

Geometrically a cube is a box where all three edges are equal. A box that is $4$ cm on each side has volume $4 \times 4 \times 4 = 64 \text{ cm}^3$. Doubling the edge turns volume from $64$ to $8 \times 64 = 512$. That is why a giant ice cube melts much more slowly than a small one , its volume grew much faster than its surface area.

There is a beautiful staircase pattern for cubes. Look at the running sums of the first few odd numbers, but this time grouped:

$1 = 1 = 1^3, \quad 3 + 5 = 8 = 2^3, \quad 7 + 9 + 11 = 27 = 3^3, \quad 13 + 15 + 17 + 19 = 64 = 4^3.$

The cube $n^3$ is the sum of $n$ consecutive odd numbers, starting just after where the previous group stopped. This was known in ancient India.

Unlike squares, the last digit of a cube can be anything: $0,1,2,3,4,5,6,7,8,9$ are all possible. In fact each last digit comes from exactly one digit of the original:

So if you know a number is a cube, its last digit instantly reveals the last digit of its cube root. The cube root of $\dots 7$ ends in $3$; the cube root of $\dots 2$ ends in $8$. Useful!

The most famous cube story belongs to the Indian mathematician Srinivasa Ramanujan. When G. H. Hardy visited him in hospital and remarked that his taxi's number $1729$ was rather dull, Ramanujan replied that $1729$ is in fact the smallest number expressible as a sum of two cubes in two different ways:

$1729 = 1^3 + 12^3 = 9^3 + 10^3.$

It has been called the Hardy–Ramanujan number ever since.

Worked examples

Example 1. Compute $7^3$.

• $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$.

Example 2. Without computing the full cube, what is the last digit of $24^3$?

• The last digit of $24$ is $4$.
• $4^3 = 64$, ending in $4$.
• So $24^3$ ends in $4$. (And indeed $24^3 = 13824$.)

Example 3. Show that $216$ is a cube by writing it as a sum of consecutive odd numbers.

• $216 = 6^3$, so it should be the sum of $6$ consecutive odd numbers.
• Start where the $5$-group ended: $5^3 = 21+23+25+27+29 = 125$. The next odd number is $31$.
• $31+33+35+37+39+41 = 216$. ✓

Example 4. Express $1729$ as a sum of two cubes in two different ways.

• $1^3 + 12^3 = 1 + 1728 = 1729$. ✓
• $9^3 + 10^3 = 729 + 1000 = 1729$. ✓

Try it yourself

1. Find $9^3$ and $11^3$.
2. Without computing, find the last digit of $37^3$ and $58^3$.
3. Is $1000$ a perfect cube? If so, of what?
4. Write $5^3$ as a sum of $5$ consecutive odd numbers.
5. List the cubes from $1^3$ to $10^3$ and note any pattern in their differences.
6. Doubling the side of a cube multiplies its volume by what factor?
7. Find another number that, like $1729$, can be written as a sum of two positive cubes in two ways. (Hint: it is bigger and starts with $4$.)
8. A cubical tank has volume $2197 \text{ litres}$. Find its edge in dm.

Activity

Build the staircase. Cut $55$ small squares of card (about $2$ cm each). Lay them out as bent strips: $1$ square, then $3$, then $5$, etc. Group the strips by the rule above (one strip, two strips, three strips, ...) and stack each group into a little cube. Count the cubies in each stack , they should be $1, 8, 27, 64, 125$. You have just seen the ancient identity that turns odd numbers into cubes.