Error Correcting Codes
Getting the Message
Activity Instructions
Purpose
This activity is designed to provide a simple way of explaining and solving the problem of data loss and
error correction. It will demonstrate how to transmit a simple code between two people even when a
third person is actively trying to block the message being sent.
Overview
There are three parts to this lesson:
Introduction
Sets up the problem of random data loss and error correction.
Thinking through the problem
Highlights the issues by working through two simple methods, neither of which can be depended on in
every scenario, logically leading up to an effective and reliable method, the Sudoku.
Activity
Opportunity for students to embed their learning and test the method using a simplified version of the
Sudoku in a group activity.
Lesson planning
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Watch the related videos on the National Literacy and Numeracy Week website.
Consider how you will work through the three lesson parts. Some teachers may like to show one or
more of the videos from the National Literacy and Numeracy Week website as a lead into the
activity, others may prefer to work through the examples from the Thinking through the Problem
document. It is important to provide some understanding for the problem by starting with some
simpler methods. These are early historical attempts at error correction which on the surface, seem
like they should work. This frames the activity and the effectiveness of the method, despite the
random errors.
Tailor the activity to your class, there are notes on difficulty level and extension included at the end
of this document.
Decide how you would like to set up the activity. Some teachers may like to include a scenario that
suits their class group or follow the activity as it is. It could be as simple as accessing your favourite
app via your phone or a secret that could save the world!
Print off the required templates listed in the equipment list
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Equipment list
All documents and templates are available on the Getting the message webpage.
Introduction – provided in this document.
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Thinking through the problem– resources developed from the Thinking through the Problem
document. Or related videos from the National Literacy and Numeracy Week website and any
equipment needed to show them, such as the IT and screen.
Printouts of 4x4 Sudoku Templates (there are two blank grids per sheet, two needed per group of
students).
A list of 120 Four-digit Passcodes with Unique Solutions. You might like to display this onscreen or
print off enough copies to cut in sections and provide a section to each group.
If required for the extension activity the Extension Activity Guide.
Optional
If there are concerns about being able to solve any of the 4 x 4 sudokus used in this activity there are
guides:
• a 4x4 Sudoku Step-by-Step Solution guide. This solves a sudoku for a specific passcode
specifying which rules are used along the way.
• a Detailed Unique Solutions guide that provides detailed solutions for every unique passcode
used in the activity.
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Lesson introduction
Setting the scene:
When you send a message on your phone, how does it get to the other end? When information is
being sent to access an app or your bank account or even stream a song, how does it get through?
The world is a noisy place. Whenever you share photos or stream a song, there are long ribbons of 0s
and 1s (the language of computers) flying around connecting you to what you want. But when static
interference occurs, the ribbon snaps. The continuous transmission of 0s and 1s becomes littered with
errors, missing or incorrect digits, making the original unreadable.
Random data loss is a problem that modern technology grapples with every day and without a solution,
our world would grind to a halt. Yet mathematicians have come up with an answer.
Your messages are converted into digital data and then protected with a buffer of excess data that you
can afford to lose. But for the buffer to work, it must provide meaning or context to the core data for the
original message to be retrievable. For example, consider the musical notation mnemonic of Every
Good Boy Deserves Fruit. The core data are the notes E G B D F and the buffer data are the words
attached to each letter. Then if some letters are randomly lost - _very G_od B_y _es_rv_s _r_it - we
are still able to work out what the words were meant to be because of the context they are in.
So, even when static interference attacks at random, there will still be a way to recover your original
message. There are various ways to create a data buffer, some more effective than others, but each
method must be written as a list of instructions: something that a computer can easily follow. These
instructions are also called an algorithm or a programme or simply, a code. As these codes correct
errors they are called Error Correcting Codes.
Thinking through the problem
What does this problem look like in practice?
Let’s start with a simplified situation. We need to access our favourite app or a bank account with a
four-digit passcode - ‘3214’ for example. For the passcode to be verified, it is sent off to a central
computer somewhere else. The problem is, when the message is sent we could randomly lose one of
the digits through static interference. This may not happen all the time or you may lose more than
one digit, but for simplicity, the activity works on the assumption that you always lose one in four
digits.
From this simple starting point, two methods are introduced representing the most straightforward
solutions possible. The shortcomings of each method are used to highlight the extent of the problem to
overcome. This frames the sudoku activity as a method that works, no matter what pattern of random
errors.
Note: To work through the examples, use the Thinking through the Problem document as a guide or
use the Getting the message – a break in transmission video available on the National Literacy and
Numeracy Week website.
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Lesson steps for the activity
1. Ideally, the class is broken up into groups of three students for the activity.
2. Explain the role of each person in the group of three:
In each group there are three roles:
The SENDER
The person who picks the
passcode and is trying to relay
it to the RECEIVER.
The JAMMER
The RECEIVER
The person who interrupts the The person who has to recover
passcode from the SENDER to the SENDER’s passcode with
the RECEIVER with errors.
the JAMMER’s errors.
The SENDER choses a four-digit passcode from the list of 120 Four-digit Passcodes with Unique
Solutions.
Each passcode is a combination of four digits using the numbers 1,2,3 or 4. Codes with repeated digits
are included in the list e.g. 1123
This four-digit passcode is entered into the 4x4 Sudoku Template.
The SENDER solves the 4x4 sudoku and passes it on to the JAMMER.
The JAMMER re-writes the completed 4x4 sudoku, leaving out up to 6 numbers of their choice,
including the original passcode.
The JAMMER passes the incomplete 4x4 sudoku to the RECEIVER who solves it to reveal the original
four-digit passcode.
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Detail
No. Steps
3
Students decide which role
they will play: the SENDER,
the RECEIVER or the
JAMMER.
4
Provide each group with two
empty 4x4 Sudoku Templates.
5
Explain to the class that the
SENDER needs to select a
passcode from the list of 120
Four-digit Passcodes with
Unique Solutions. The code is to
be written in the code squares
which are patterned boxes above
the top of each 4x4 sudoku.
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Illustration/Example
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No. Steps
6
Illustration/Example
The SENDER then enters each
digit of the passcode into the
empty 4x4 sudoku by matching
the pattern of the code square to
the respective pattern in the grid.
The SENDER then completes
the template following rules
similar to a regular 9x9 sudoku.
The rules are specified in the
following steps.
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Numbers 1 through to 4 appear
once in every row…
8
…once in every column…
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No. Steps
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…and once in every 2x2 block.
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Following the rules of 4x4
sudoku the SENDER completes
the grid.
Illustration/Example
If you have trouble solving the
sudoku there is a 4x4 Sudoku
Step-by-Step Solution document
to use as a guide.
The completed grid is then given
to the JAMMER.
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Explain to the class that the
JAMMER rewrites numbers from
the completed sudoku grid
leaving empty all the code
squares (patterned squares in
the grid). He or she can remove
up to six numbers in total from
the SENDER’s completed grid.
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No. Steps
12
The JAMMER then gives only
the incomplete sudoku grid to the
RECIEVER.
Explain to the class that the
JAMMER now needs to complete
the sudoku to find the passcode.
13
Once the sudoku has been
correctly filled in there will be
numbers in the patterned code
squares.
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Explain to the class that the
JAMMER can now fill in the code
squares at the top of the sudoku
grid which correspond to the
patterned code squares in the
grid.
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Illustration/Example
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No. Steps
15
Illustration/Example
The RECEIVER then checks with
the SENDER if the code has
been correctly recovered. If so,
ACCESS GRANTED and if not…
THE JAMMER HAS WON!
Additional Teachers Notes
Passcodes in the list of 120 Fourdigit Passcodes with Unique
Solutions are arranged numerically
from smallest to biggest.
They are also split into 5 columns: A,
B, C, D and E.
These columns refer to separate
documents where each passcode is
given a detailed solution in the 4x4
Sudoku. The page number next to
each passcode refers to the location
of the solution in the respective
document. The 5 documents are:
Detailed Unique Solutions_Column A
Detailed Unique Solutions_Column B
Detailed Unique Solutions_Column C
Detailed Unique Solutions_Column D
Detailed Unique Solutions_Column E
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Difficulty
This activity is designed to be as straightforward as possible.
As such, we have limited the number of four-digit passcodes
that can be used. Each code in the list of 120 Four-digit
Passcodes with Unique Solutions only has one solution when
entered in our 4x4 sudoku grid. This is not true for every
combination code. For example, the code 3333 entered in our
4x4 sudoku grid can be solved 18 different ways.
Without any restrictions there are a total of 256 different four
digit codes using 1,2,3 or 4. There are many of these codes
that could also be used in a 4x4 sudoku but they no longer
produce a unique grid. There are also some codes that do not
work at all in a 4x4 sudoku. We leave open the use of these
more exotic passcodes as an extension activity for the more
adventurous sudoku masterminds. For details see the
Extension Activity Guide.
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