Number Bracelets

Introduction

Here are the first three numbers of a number bracelet starting with 1:

1, 3, 4, ... can you deduce what the next number is?

It's 7! So the sequence so far is 1, 3, 4, 7, ... any thought what the next one is? What is your rule?

It's 1! So the sequence so far is 1, 3, 4, 7, 1, ... did you get the right number? What do you think the rule is?

Watch this video for an explanation of how to make a Number Bracelet.

You must start by choosing two whole numbers from 0 to 9, and then add those two to find the next number in the "bracelet". Follow the instructions in the video until you have a good mathematical reason to stop.

Exploring Number Bracelets

Now try starting with a different pair of numbers. Remember, you should choose two numbers from 0 to 9, and it's okay to choose the same number twice.

Explore Number Bracelets with different starting numbers to answer the questions below.

1. What is the longest Number Bracelet that you can find?

2. What is the shortest?

3. How many different pairs of starting numbers are there? (Hint: think of the starting pair as a two-digit number, so if you start with '3' and '2', that would be '32').

4. Does each pair of starting numbers give a different Number Bracelet?

5. How many different Number Bracelets are there?

6. How do you know when you have found them all?

Further Questions and Challenges

Now that you have found all the different Number Bracelets, let's explore them further.

1. For each Number Bracelet, how many numbers are there before it starts to repeat? How many of those are odd numbers, and how many are even numbers?

2. Explore the sequence of odds and evens in each Number Bracelet (you may want to use these templates to help you). What do you notice?

3. Fill in this grid, using a different colour for each of the different Number Bracelets. What patterns can you spot? This video will help you to understand what to do.

4. For this task you should work with the longest Number Bracelet. Download one of these grids and fill in the numbers for the Number Bracelet, in order, one row at a time. When you have finished, see what patterns you can spot in the columns and the diagonals of the grid. Try it with different sized grids. What d you notice? This video will help you to understand the task.

Other Types of Number Chains

There are lots of different number chains that can be made by choosing two starting numbers, and then using a rule to work out the next number. Not all of them lead to "bracelets", i.e. patterns that repeat after a while. Here are some other number chains that you might like to epxlore to see what happens with them, and to try to understand why.

1. Use the same idea as for the Number Bracelets in this task, but this time multiply instead of adding the two numbers.

2. Choose any two-digit number to start with, e.g. 47. Multiply the "ones" digit (in this case 7) by 3, and add it to the "tens" digit (in this case 4). This gives 7 x 3 + 4 = 25, so 25 is the next number in the chain. Continue applying the same rule. What happens? Choose a different starting number and see what happens then. Repeat with a few more numbers and try to explain what is happening.

3. Choose any number to start with (but it's best to choose one that's less that 100 to make the calcualtions easier, at least to start with). If the number is even, divide it by 2; if the number is odd, multiply it by and then add 1. For example, if you start with 10, the next number would be 5 (it's even, so divide by 2). After 5 it would be 16 (5 is odd, so multiply 5 by 3 and add 1). What happens? How long is each chain? Do any numbers not behave in this way?

Further Practice

Some essential skills introduced in this lesson are addition and subtraction. Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.

Transum

• Formal Written Methods - examples of formal written methods for addition, subtraction, multiplication and division.

• Number Skills Inventory - a checklist of basic numeracy techniques that every pupil should know. Look especially at the adding and subtracting activities.

• Mix and Math - determine the nature of adding, subtracting and multiplying numbers with specific properties - not everything here is relevant.

• What Are They? - an online exercise about sums, products, differences, ratios, square and prime numbers - not everything here is relevant.

Extension

Transum

Try the following puzzles:

• No Partner: Find which numbers in a given list do not combine with other numbers on the list to make a given sum.

• Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals

• Suko Sujiko: Interactive number-based logic puzzles similar to those featuring in daily newspapers.

• Magic Square Puzzle: Find all of the possible ways of making the magic total from the numbers in this four by four magic square.