Keeping score of a game

Exploring the maths behind keeping score as players work at the game of Fifty Wunners

This task concerns only the scoring aspects of a game. I do not know how the game works ~ it can be played by any number of people.

Scoreboards for the game
Scoreboards and game ready to start

In this game, players take turns to 'do something'. This project is not concerned with what the 'something' is. It is only interested in the scoring system.

At the end of each turn, a player will score 1 ~ or 3 ~ or 5 points. Each player keeps a record of his or her total points on a personal scoreboard.

Each board has three sections ~ they are marked Store ~ Wunners ~ Tenners. At the start of the game each player has fourteen counters in the Store Section.

Counters on the scoreboard
Counters on the scoreboard ~ start of game

Players take it in turns to play. They keep their individual scores by moving 1 ~ or 3 ~ or 5 ~ counters from the Store to the Wunners Section.

Scoring in progress
Whose turn is it next?

At any stage of the game ~ usually after a turn ~ it is possibe to return a bundle of ten counters from the Wunners Section back to Store, and exchange them for a single counter from Store that must be placed in the Tenners Section. One counter in the Tenners Section is equivalent to ten from the Wunners.

The winner is the first player to reach a score of fifty points or more ~ busy doing their 'Somethings'.

Here is a sample for a recent game which has been in progress for a while.

Scoreboard showing various points
Which player has the highest score? ~
Which player finds the game most difficult? ~
A turn is about to be completed by ~ which player?

Let us consider a game ~ round-by-round ~ from the start.

After the first round, there are three possible scores that a player could have. What are they?

I suggest that you make some notes at this point ~ ready for the next stage. It really is important to think carefully about possible scores.

Can you invent a system for keeping a record of what is going on? It would be a great help for you with the next few questions.

After the second round, what are the possible scores that one player could have?

Scoreboards after two rounds
Possible scoreboards after two rounds

Think about the next question and make some notes. There are lots of possibilities ~ maybe too many to remember all at once. I find writing very helpful.

After the third round, what possible scores could players have?

When you have thought about ~ and answered ~ the theoretical question ~ above ~ follow the game as it has developed ~ below.

Scoreboards after three rounds ~ with an error
Scoreboards after three rounds. There is an error.
Which player has made a scoring mistake?

Let's go back to some theoretical questions. What are the answers to these?

What is the least number of rounds that can be played?
What is the greatest number of rounds that can be played?
How many rounds do you think the average game might last?

If you have been thinking mathematically about earlier work, you will have been able to notice that one player needs some supervision.

In the scoreboards picture above ~ how can you tell which player has made a scoring mistake?

Now back to the game. There has been another round. I think trouble is brewing. After the fourth round the scores look like this ~

Scoreboards after four rounds error compounded
Scoreboards after four rounds ~
Has the previous error been corrected? ~
Has it got worse?

It is best that we stop following that game now. One player has gained an incorrect advantage ~ you may have spotted it. Future games will have to be played with a good umpire to keep check.

Another ~ different ~ game progressed very well. It looked like this as players approached the finish ~

Scoreboards near the end of the game
Scoreboards near the end of a game

After a number of rounds the players were invited to make predictions about their position as the end of the game was in sight ~ remember, the aim is to be the first to fifty ~ or more.

North says 'I shall win next round'
East says 'I might win in two rounds'
South says 'I might win next round'
West says 'I cannot win in the next three rounds'

What do you say? Jot down your thoughts ~ ready to discuss with your tutor.

I say 'North and East have equal chances.' Am I right or wrong?

'I say 'West is quite right ~ sadly ~ no chance.' Am I right or wrong?

The ages of the four players ~ not in any particular order ~ are 9~8~6~9. Do the results reflect the average age of the players?

For some students only ~ Discuss the meanings that might be placed on the word 'average' in the previous sentence.

Some ideas for developing this task further are available ~ without subscription [115030].

Answers and commentary are available for BusyBusy subscribers [114503].

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