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Out for the count: the mathematics of voting systems

Updated Monday, 14 February 2022
Voters, voters, in the poll, which is the fairest system of all?

Democracy relies on fair and free elections, but how does the voting system affect the outcome? Across the UK and the world, different systems are used: the Additional Member System for the Scottish Parliament and Welsh Senedd, the Single Transferable Vote for the Northern Irish Assembly, the Alternative Vote for Irish presidential elections, and First Past The Post for Westminster.

PDF document Transcript 80.1 KB

But which is the fairest and/or most effective? In this hands-on workshop, you will discover how maths can help us decide what we mean by the “fairness” of a voting system. We will be counting ballots in a mock election. Watch the video below to get started!

PDF document Transcript 56.1 KB

To participate fully in this workshop, you will need to engage with the ballot count activity. You have two options for accessing the ballots, as laid out below.


Digital ballots

Click and drag the ballots to wherever you want on the screen, and even stack them on top of one another.

Once you are all set up, proceed on to watch the next video!

Print out paper ballots

If you have access to a printer and a pair of scissors, you can print out the ballots and cut them out. There should be 11 ballots in total, as well as headers marked “A”, “B”, “C” and “D”, which you can use to help keep track of your piles of counted votes.

To download our PDF to print out, click here. (PDF document146.1 KB)  

Once you have printed out and cut out your ballots, proceed on to watch the next video!


  Transcript (PDF document101.3 KB)

Time to count! Use the First Past The Post system to count the ballots and declare a winner! Remember to count only the first-preference votes.

PDF document Transcript 84.1 KB

Time to count! Use the Alternative Vote system to count the ballots and declare a winner. You might need to watch Video 3 again to remind yourself how to deal with a tie.

Once you have declared a winner, reflect on whether you thought this system was fair or unfair.

  Transcript (PDF document83.0 KB)

Time to count! Use the Borda system to count the ballots and declare a winner. Remember the points are awarded as follows:

  • 4 points for a first-preference vote
  • 3 points for a second-preference vote
  • 2 points for a third-preference vote
  • 1 point for a fourth-preference vote

You might like to use a notepad to keep a tally of the votes as you count them:

Transcript (PDF document86.0 KB)

Time to count! Use the Condorcet pairwise-counting system to count the ballots and declare a winner. Remember, you will need to count the ballots for each pair of candidates pitted against one another.

You might like to use a notepad to keep track of the pairs as you count them: 

  Transcript (PDF document81.3 KB)

Thank you so much for taking part in this activity! We hope that you have enjoyed it, and that it has helped you understand how mathematics can help us to understand concepts of fairness that help us make informed choices about our democracy.   

Acknowledgements: This learning activity was adapted from materials purchased under license from Outstanding Resources, and is provided here solely for the purpose of your own private study. For use in teaching in an educational setting, you must purchase a single-user license from the Times Educational Supplement website. And a special thanks to Rebecca Rosenberg, Lecturer in Mathematics Education, for her help in developing this activity.


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