Srinivasa Ramanujan and number patterns

Quick answer

Srinivasa Ramanujan was a brilliant Indian mathematician who discovered amazing number patterns, including the famous "taxicab number" 1729, which is the smallest number that can be written as the sum of two cubes in two different ways.

Learning outcome

By the end of this page, you will understand how to spot number patterns like Ramanujan did, and you will be able to find numbers that can be written as the sum of two cubes in different ways.

• You can identify numbers that are the sum of two cubes in two different ways
• You can create your own number patterns using cubes
• You can use addition and multiplication to solve puzzles about number patterns
• You can share your discoveries about number patterns with others

Why this matters

Number patterns are everywhere in our world! From the way flowers grow, to how we count our money, to how games are designed, patterns help us understand and predict what comes next. Ramanujan was a genius who saw patterns that others couldn't, and his work still helps mathematicians today. Learning to spot patterns helps you become better at maths, and it makes you more curious about the world around you.

How it works

Let's start by understanding what cubes are. A cube is a number multiplied by itself three times. For example:

1. 1 cubed = 1 × 1 × 1 = 1
2. 2 cubed = 2 × 2 × 2 = 8
3. 3 cubed = 3 × 3 × 3 = 27
4. 4 cubed = 4 × 4 × 4 = 64
5. 5 cubed = 5 × 5 × 5 = 125

Now, we want to find numbers that can be made by adding two cubes together in two different ways. For example, we want to find a number that equals:

• A cubed + B cubed
• C cubed + D cubed

Where A, B, C, and D are different numbers.

Examples you can see

Let's look at a fun example with small numbers. We'll try to find a number that can be written as the sum of two cubes in two different ways.

First, let's list the cubes we know:

Now, let's try to add two cubes together and see if we can make the same number in two different ways.

Let's try 1729:

• 1 cubed + 12 cubed = 1 + 1728 = 1729
• 9 cubed + 10 cubed = 729 + 1000 = 1729

So 1729 is the smallest number that can be written as the sum of two cubes in two different ways!

Let's try another one:

• 1 cubed + 10 cubed = 1 + 1000 = 1001
• 7 cubed + 8 cubed = 343 + 512 = 855

These don't match, so 1001 is not the answer. Let's try 1331:

• 1 cubed + 12 cubed = 1 + 1728 = 1729
• 9 cubed + 10 cubed = 729 + 1000 = 1729

These are the same numbers again. Let's try 855:

• 1 cubed + 9 cubed = 1 + 729 = 730
• 2 cubed + 8 cubed = 8 + 512 = 520

This doesn't work either. Let's look for 1729 again because it's special.

Try it as a mini adventure

Imagine you are a detective trying to solve a mystery about numbers! You've been given a clue that there is a special number called the "taxicab number". This number is the smallest number that can be written as the sum of two cubes in two different ways.

Your mission is to find this number! Here's your detective kit:

1. Write down all the cubes from 1 to 10 (you already did this!)
2. Try adding pairs of cubes together
3. Look for the same answer twice
4. When you find it, write down the two ways you found it

Let's try a few combinations:

• 1 + 8 = 9
• 1 + 27 = 28
• 8 + 27 = 35
• 1 + 64 = 65
• 8 + 64 = 72
• 27 + 64 = 91
• 1 + 125 = 126
• 8 + 125 = 133
• 27 + 125 = 152
• 64 + 125 = 189

Keep going until you find the number that appears twice!

4th time: 1 + 1728 = 1729

5th time: 729 + 1000 = 1729

Congratulations! You've found the taxicab number 1729. You're a number detective!

Common mistakes

• Mistake: Thinking that 1729 is the only number that can be written as the sum of two cubes in two different ways.
• Why it happens: Children might think that because 1729 is special, it's the only one.
• Correct answer: There are other numbers with this property, but 1729 is the smallest.

• Mistake: Forgetting that 1 cubed = 1 and not 1.
• Why it happens: When multiplying numbers, it's easy to forget that 1 × 1 × 1 = 1.
• Correct answer: Always remember that 1 cubed = 1.

• Mistake: Adding cubes incorrectly.
• Why it happens: When doing calculations, children might make simple addition errors.
• Correct answer: Double-check your calculations. Use a calculator or count carefully.

Parent and carer guide

Helping your child explore number patterns can be a fun and rewarding experience. Here are some tips to support your child's learning:

• Encourage your child to write down the cubes of numbers from 1 to 10, and then add them in pairs.
• Ask your child to explain how they found 1729. This helps them understand their own thinking.
• Use real-world examples like counting how many legs different animals have, or how many wheels are on a bicycle, to help them understand patterns.
• Don't worry if your child gets stuck. It's normal to take time to find patterns. Encourage them to keep trying.
• Make it a game! You could make a chart and see who can find the most number patterns in a set time.

Remember, this is about exploration and discovery. It's okay if your child doesn't get it right away. The joy is in the journey of finding patterns and making connections.

Practice questions

1. What is 3 cubed?
2. What is 4 cubed?
3. Find a number that can be written as the sum of two cubes in two different ways.
4. Is 1000 a cube? Why or why not?
5. What is 1729 minus 1?

Answers:

1. 3 cubed = 3 × 3 × 3 = 27
2. 4 cubed = 4 × 4 × 4 = 64
3. 1729 is the answer (1³ + 12³ = 1729 and 9³ + 10³ = 1729)
4. Yes, 1000 is 10 cubed (10 × 10 × 10 = 1000)
5. 1729 - 1 = 1728

FAQ

Who was Srinivasa Ramanujan?

Srinivasa Ramanujan was an Indian mathematician who lived from 1887 to 1920. He was very good at spotting number patterns and solving difficult maths problems. Even though he had little formal training, he made many important discoveries in mathematics.

Why is 1729 special?

1729 is special because it's the smallest number that can be written as the sum of two cubes in two different ways. This makes it a very interesting number for mathematicians.

How can I find more number patterns?

You can find more patterns by looking at numbers in different ways. Try adding, subtracting, multiplying, or dividing them. Look for repeated numbers, or numbers that follow a rule. You can also look at patterns in nature, like the number of petals on flowers or the number of sides on shapes.

What are cubes in maths?

Cubes in maths are numbers that are multiplied by themselves three times. For example, 2 cubed is 2 × 2 × 2 = 8.

Can I use this in games?

Yes! You can make games where you find numbers that fit certain patterns. You can also use cubes to build patterns with blocks or shapes. This helps you understand how numbers work together.