First connect three
Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?
Age
7 to 11
Challenge level
Problem
First Connect Three printable sheet
In this game the winner is the first to complete a row of three, either horizontally, vertically or diagonally.
Spin the spinners, place each number in one of the squares and decide whether you want to add or subtract to produce a total shown on the board. Your total will then be covered with a counter.
You cannot cover a number which has already been covered.
If you are unable to find a total which has not been covered you must Pass.
You can use the interactive version below or use two 1-6 dice to play away from the computer.
Are there some numbers that we should be aiming for? Why?
Which number on the grid is the easiest to get? Why?
Which number is the most difficult to get? Why?
Printable NRICH Roadshow resources: Instructions and Game board.
For a more challenging version of this game, you could look at Connect Three, and/or change the settings of the interactivity by clicking on the purple cog.
Getting Started
Are some numbers part of more lines of three than others? This might affect whether we aim for them or not.
How can you get 1 by spinning the spinners (or by throwing two standard 1-6 dice)? How many ways are there altogether?
How can you get 12 by spinning the spinners (or by throwing two standard 1-6 dice)?
It might be useful to look at all the numbers on the grid in turn in this way.
Student Solutions
Daisy and Izzy from Thomas Eaton Primary Academy described a similar strategy to each other. Daisy wrote:
First I tried to make the highest number I could. Then with the numbers I got I would make the closest possible answer to go near it. When ever my opponent was getting close to winning I would focus on getting in their way so they wouldn't win.
Izzy added:
Then you need to keep an eye on the person you're playing against as they will block you. Try get three in a row but keep your options open as you may not get the numbers you wanted.
Sara from the International School of Brussels shared her work:
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%202019-10-15%2009-59-41.jpg "First Connect Three")
and this came from Smilla, also from the International School of Brussels:
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Adam from Willowbank in New Zealand thought it was best to start in a different place:
I think it's best to try and aim for the 6 numbers in the middle 0,1,2,5,6,7 because you have a high chance of winning because it will be easy to get a diagonal horizontal or vertical 3.
Coen from Bainbridge School in the UK used a similar strategy, but for slightly different reasons:
If I was you, I would aim for the lower numbers because you have got more of a chance to win. The easiest number to get is 7 because you quite often get a 3 or a 4, but if you takeaway then you will get 1.
The minus numbers are the hardest numbers to get because you quite often get higher numbers and if you takeaway you will probably get 1, 2, 3 or 4. But if you get 5, 6 or 7 when you have rolled the dice and the other one says 5 then you could possibly get -1, -2, 0.
Louisa from Walton High School in the UK also thought about which would be the easiest and hardest numbers to get:
The hardest numbers to get are -5 and 12 because for both of them there is only one combination of numbers which you can roll to get them as a total.
Daniella, Sandra and Michelle from Park Lane International School in the Czech Republic worked out how many ways there are of making each number, and how many lines there are through each number:
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Caleb from Kilvington in Australia used this idea to work out which lines connecting 3 numbers are most likely to be the winning lines:
The most likely 3 in-a-rows you can get are: 5,6,7 ; 6,7,8 ; -1,0,1 ; 0,1,2.
The most likely 3 in-a-diagonals you can get are: 1,5,9 ; -2,2,6.
The most likely 3 in-a-columns you can get are: -2,3,8 ; -3,2,7 ; -1,4,9, ; 0,5,10.
All of these methods use at least two of the numbers from -1 to 8. Also, the methods that are COMPLETELY made up of numbers from -1 to 8 give you the highest probability of winning. This is why I believe that in this game you must aim for -1 to 8.
Teachers' Resources
Why do this problem?
This problem is a great way for students to take responsibility for their own learning. They can avoid negative numbers if they are not confident or they can push themselves to calculate negative answers. In analysing the game more fully, rather than just playing it, the idea is that learners can develop a system for finding all the possible ways of making each number on the grid, so they can justify which are the easiest to get.
Possible approach
This printable worksheet may be useful: First Connect Three.
You could introduce the game by playing against the class, or by splitting the class into two teams to play against each other, or with the class playing against the computer. Students can play against each other in pairs to get more of an idea of the game. You can print off this board if the students are not playing at a computer or laptop and use dice instead of the interactive spinners.
After a suitable length of time, ask the suggested questions in a whole-class discussion that focuses on emerging strategies, observations, explanations and justifications. Students can then go back to working in pairs to establish the numbers of ways of achieving the different totals.
At the end of the lesson a plenary discussion can offer a chance to present findings and you can draw attention to those methods which were particularly efficient. This would then lead to a discussion about how their findings might affect the way they play the game to win.
Key questions
Are there some numbers that we should be aiming for? Why?
Which numbers on the grid are the easiest to get? Why?
Which numbers are most difficult to get? Why?
Possible extension
Further challenges could be provided by asking what would happen if:
- there was a differently shaped board
- numbers appeared more than once on the board and you could place more than one counter in a turn
- you could use 1-12 spinners (or dice)
- you wanted to design a board for a game where you allowed multiplication and division
For students who are able to add and subtract both positive and negative numbers, Connect Three is a suitable extension.
Possible support
If some pupils are struggling, you could adapt the board so that it only contains the numbers 1-12. You could also suggest pupils play in groups of four, one pair against another, in order to support each other while carrying out calculations.