Nim-7
Problem
This is a basic form of the ancient game of Nim.
You will need seven objects, such as counters or blocks. It is a game for two players.
Watch the video below to find out how to play:
If you cannot access the video, click below to read the rules.
Place the seven counters in a line and decide who will go first. (In the next game, the other player will have the first turn!)
Each player takes it in turns to take away either one counter or two counters.
The player who takes the last counter (or counters) wins.
Play several times so that you get a good 'feel' for the game.
Are there any points in the game, before the end, when you know who the winner is going to be? How do you know?
Can you find a way to play so that you are sure you will win right from the start?
Does it matter who has the first turn? Why or why not?
Once you are an expert at this game, you may like to try playing Daisy, which is another Nim-like game.
Printable NRICH Roadshow resource.
Getting Started
What happens when there are three counters left?
Student Solutions
We had some great ideas sent in for this task, so thank you to everybody who sent us their solutions.
Lots of children noticed that if there are three counters left at the beginning of your go, you can't win. Josh, Arthur and Mia from Kirkby on Bain in the UK said:
We have noticed that when there are three counters left no matter what the next player takes, the other player will win. This is because there will be one left if two are taken, and two left if one is taken. So the other player will win!
Josh, Arthur and Mia then explained how the first player could win:
Player 1 should take one counter as their first go. This means they will get to leave three counters after their second go and then win. We are going to try with 9 counters!
Thank you for sending in these ideas. I wonder what the best strategy for winning is with nine counters?
We also had lots of ideas sent in from the children at St Charles Primary School Ryde in Australia. Samuel suggested that if there are seven counters left, you should take one counter so that there are six left. Can you definitely win if you leave six counters for the other player? Why?
The children from Olga Primary School in the UK sent in lots of videos explaining their ideas. Francis and Caitlin explained why leaving three counters for the other player is a winning strategy:
Ben and Ayaan explained why going first means you can always win:
Thank you all for sending in your thoughts about this game!
Thank you as well to Eloan and Ella from Clifton College Prep School in England, Joshua from Spring Hill Primary School in Australia, the children from Waverley Primary School in the UK, and Dhruv from Pict in India, who all had similar strategies.
Teachers' Resources
Why play this game?
This game offers a motivating context in which children can improve their logical thinking skills. It is a low threshold high ceiling game that is easily accessible but, at the highest level, has the potential to be generalised.
Possible approach
This problem featured in an NRICH Primary webinar in June 2021.
Introduce the game to the class by watching the video all through. Invite learners to ask questions or make comments, and use these to help clarify the rules. You may want to watch all or part of the video again if there are any uncertainties. Give pupils chance to play the game several times in pairs using seven counters or any other objects, so that they get a really good 'feel' for it.
Bring everyone together and explain that you're now going to focus on 'strategy', in other words, ways to win the game. Invite learners to share anything they have noticed so far with sentences such as, "I noticed that when I ..., xxxx happened". Try to value all children's noticings and then use the video again to focus your questioning.
Play the video from the beginning again, but this time pause it after Player 1's second turn (about 37 seconds in). Ask the class what they would do now if they were Player 2. Give everyone the chance to talk to a partner about their ideas, then draw the whole group together again. What do they notice? In fact, it is impossible for Player 2 to win now. Encourage learners to articulate the reasons for this by thinking more than one step ahead, and considering all possibilities, for example "If I took one counter, then the other player would..."; "If I took two counters, then...".
Help the class to understand that if we want to win this game, we need to leave exactly three counters for our opponent to have a turn. Challenge them to consider whether it is possible to be absolutely certain that you will be able to leave three counters. Give them lots more time to play the game and explore their ideas. They may find it helpful to record their moves somehow and you can look out for useful ways to do this amongst the class and share them, as appropriate.
Any pair who thinks they have found a completely water-tight strategy can try it out in a game against you. You may also like the class to think abut how they can record their winning strategy, perhaps in the form of 'Top Tips' for a friend.
This game is a great one to share with families. Having introduced it in class, you can suggest that children teach a family member to play as part of their home learning.
Key questions
What happens when there are three counters left?
Does it matter who goes first? Why or why not?
How can you win at this game?
Possible support
You could offer to record a game for children who are struggling, and then you can then look back together at key moments. This might enable you to discuss what each player could have done differently at certain points in the game.
Playing with one pair against another pair means that learners have someone to talk to about their ideas, giving them confidence to play alone.
Possible extension
You can encourage the children to think about 'What if...?' questions, such as 'What happens if you start the game with a different number of counters?'. (A series of key numbers will emerge, as well as some interesting observations about odds, evens and multiples.) The game Got It is identical in structure to Nim-7 and would make a great follow-up task. The game Daisy, another Nim-like game, offers an interesting challenge too.
Nice or Nasty
Problem
Nice or Nasty - printable instructions sheet
Nice or Nasty - printable scoring sheet
Find a partner and a $1-6$ dice, or even a $0-9$ dice if you have one. You could use the dice in Dice and Spinners.
Each of you draw a set of four boxes like this:
Or you can download and print off this scoring sheet.
Game $1$
Take turns to roll the dice and decide which of your four boxes to fill. Do this four times each until all your boxes are full. Read the four digits as a whole number.
Whoever has the larger four-digit number wins.
There are two possible scoring systems:
- A point for a win. The first person to reach $10$ wins the game
- Work out the difference between the two four-digit numbers after each round.
The winner keeps this score. First to $10000$ wins.
Now for some variations...
Game $2$
Whoever makes the smaller four digit number wins. You'll probably want to change the scoring system.
Game $3$
Set a target to aim for. Then throw the dice four times each and work out how far each of you is from the target number. Whoever is the closer wins.
There are two possible scoring systems:
- A point for a win. The first person to reach $10$ wins the game
- Work out the difference between the two four-digit numbers and the target number after each round. Keep a running total. First to $10000$ loses.
Game $4$
This game introduces a decimal point. The decimal point will take up one of the cells so this time the dice only needs to be thrown three times by each player. Choose a target. The winner is the one closer to the target.
Two possible versions:
- each player decides in advance where they want to put the decimal point before taking turns to throw the dice
- each player throws the dice three times and then decides where to place the digits and the decimal point.
Again, two different scoring systems are possible.
Game $5$
This is the nasty version!
Play any of the games above. This time you can choose to keep your number and put it in one of your cells, OR give it to your partner and tell them which cell to put it in. You might lose a friend this way! It's really important to take turns to start each round if this game is going to be fair.
This becomes even nastier when you play the games above with more than two people.
Game $6$
A cooperative game rather than a competitive one - for three or more people.
Choose any of the games above. Decide in advance which of you will get the closest to the target, who will be second closest, third, fourth etc. Now work together to decide in whose cells the numbers should be placed, and where.
Teachers' Resources
Nice or Nasty
Find a partner and a $1-6$ dice, or even a $0-9$ dice if you have one. You could use the dice in Dice and Spinners.
Each of you draw a set of four boxes like this:
Or you can download and print off this scoring sheet.
Game $1$
Take turns to roll the dice and decide which of your four boxes to fill. Do this four times each until all your boxes are full. Read the four digits as a whole number.
Whoever has the larger four-digit number wins.
There are two possible scoring systems:
- A point for a win. The first person to reach $10$ wins the game
- Work out the difference between the two four-digit numbers after each round.
The winner keeps this score. First to $10000$ wins.
Now for some variations...
Game $2$
Whoever makes the smaller four digit number wins. You'll probably want to change the scoring system.
Game $3$
Set a target to aim for. Then throw the dice four times each and work out how far each of you is from the target number. Whoever is the closer wins.
There are two possible scoring systems:
- A point for a win. The first person to reach $10$ wins the game
- Work out the difference between the two four-digit numbers and the target number after each round. Keep a running total. First to $10000$ loses.
Game $4$
This game introduces a decimal point. The decimal point will take up one of the cells so this time the dice only needs to be thrown three times by each player. The winner is the one closer to the target. Choose a target.
Two possible versions:
- each player decides in advance where they want to put the decimal point before taking turns to throw the dice
- each player throws the dice three times and then decides where to place the digits and the decimal point.
Again, two different scoring systems are possible.
Game $5$
This is the nasty version!
Play any of the games above. This time you can choose to keep your number and put it in one of your cells, OR give it to your partner and tell them which cell to put it in. You might lose a friend this way! It's really important to take turns to start each round if this game is going to be fair.
This becomes even nastier when you play the games above with more than two people.
Game $6$
A cooperative game rather than a competitive one - for three or more people.
Choose any of the games above. Decide in advance which of you will get the closest to the target, who will be second closest, third, fourth etc. Now work together to decide in whose cells the numbers should be placed, and where.
Why play these games?
These games are thought provoking and very engaging. They encourage discussion of place value, and strategic mathematical thinking.
Resource downloads
Nice or Nasty - Scoring Sheet
Possible approach
Key questions
Possible support
Start with two, then three boxes, before moving onto four. Choose the easiest scoring system or allow calculators for scoring the more difficult version. Allow pairs of students to play against other pairs, so that they can support each other.
Possible extension
You may wish to move the students on to Dicey Operations in Line or Dicey Operations (the former might be more suitable for primary learners; the latter for secondary students).
Factors and Multiples Game
Factors and Multiples printable worksheet - game
Factors and Multiples printable worksheet - puzzle
On this resources page there are printable 1-100 square grids.
You may like to have a go at Dozens before having a go at this challenge.
This is a game for two players.
The first player chooses a positive even number that is less than 50, and crosses it out on the grid.
The second player chooses a number to cross out. The number must be a factor or multiple of the first number.
Players continue to take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player.
The first person who is unable to cross out a number loses.
Play a few times to get a feel for the game.
Do you have any winning strategies?
Here is an interactive version of the game in which you drag the numbers from the left hand grid and drop them on the right hand grid. Alternatively, click on a number in the left hand grid and it will transport to the earliest empty location in the right hand grid. You can rearrange the numbers in the right hand grid by dragging and dropping them in position. The integer in the top right hand corner grows with the number of factors/multiples you have in a row.
Tablet version Install in home page
This activity featured in an NRICH video in June 2020 and an NRICH student webinar in November 2020.
You may like to have a go at Dozens before having a go at this challenge.
Many thanks to everyone who shared their solutions with our team. We were very impressed by your resilience and perseverence to produce the chains that you submitted to NRICH. Several of you sent in more than one solution, substituting an earlier chain with an even longer one after spending more time on the challenge. Well done to you all!
We've published below a selection of increasingly long chains which have a screenshot showing the full chain, or a list of its factors and multiples. We hope this list inspires you to keep working on your own solutions and perhaps consider the length of the longest possible chain, and the reasons for your answer.
Let's begin with Montsaye Community College in Northamptonshire where their students took on the challenge of generating the longest possible chain.
Gabrielle and Lauren reached a total of 50 numbers:
Aravindan, from GHS in India, shared this chain of 51 numbers:
Makenzie, from Mountsaye Community College, managed to improve on this with a chain of 55 numbers:
Sophie and Tasmin, also from Mountsaye, managed to improve on that with this chain of 56 numbers:
Then, Gabrielle and Lauren managed to improve on their earlier effort with this new chain featuring 59 numbers:
Aravindan improved on his previous effort to produce this chain of 60 numbers:
Abigail from Ridgewood School managed a chain of 61 numbers:
Alfie, Manuel, Jack and Emilio from Newhall School in Chelmsford, Essex, worked as a team to also produce a chain of 61, and Harry from Farmborough Church Primary School created the same chain:
A.H. from Manorfield Primary School has improved on this by finding a chain of 63 numbers:
90-9-99-33-66-11-44-22-88-8-80-40-10-100-20-60-30-15-75-25-50-5-35-7-70-14-56-28-84-21-42-6-78-39-13-26-52-4-68-34-17-51-1-46-92-23-69-3-57-19-38-76-2-24-72-18-36-12-48-16-24-32-96
This solution came in from Ralph and Max at the Institut International de Lancy in Switzerland. Their teacher said they did not use a computer or calculator. They made a chain of 65, which seems to be a very popular answer.
Evie from Deansfield Primary School created a chain of 65 numbers:
James from Ridgewood School went one better with this chain of 66 numbers:
Linda from Bohunt School used 68 numbers:
A group of Year 9 students from The Perse School for Girls in Cambridge worked together and managed an even longer chain of 71 numbers:
And Claire, of Blackheath High School in London, has managed to improve on that with this chain of 73 numbers:
Jacky, from Princethorpe College in Rugby, has managed to go one better with a chain of 74:
Jesse from Moriah College in Sydney, Australia also managed a chain of 74 numbers:
Izaak, from Hills Rd VI Form College, used some computer programming techniques to try to make the longest chain he could. Below is his 76 long chain. If you want to read about his program and see his code, he has shared it on Github.
We've also received a few chains using 77 numbers!
Millie from Hursthead Junior School in the UK sent in this solution:
Well done Millie! Jakob from Haberdashers Adams School in the UK also found the same solution.
We received a different chain of 77 numbers from Michelle at The Coopers' Company and Coborn School in the UK:
Ahaan from Newcastle Under Lyme School found the same chain, and Sarisha from Newcastle Under Lyme School found this chain of 77 numbers:
Can you see how this is related to Michelle's chain above?
Miraya from Heckmondwike Grammar School shared these strategies for developing longer and longer chains:
First of all not use the numbers one to ten since they are the common multiples or factors of any given number between one to hundred. Next , the other thing you have to remember is to try not to use prime numbers as the prime numbers have factors that are only 1 you can use multiples but they will not last very long. Try to find out the common multiple factors as well. Also, try to use the bigger numbers since they will have the most factors. Finally , try to use the common numbers with the most multiples under 100 for example 10 or 11 since they have multiples in every ten numbers.
Well done to all of you, and to those of you who found chains of 77, congratulations, you have found the longest chain.
Thank you to Josh Tutin who sent us a link to The On-Line Encyclopedia of Integer Sequences, and the table below, which lists the longest possible chains using the numbers from 1 to n, where n is smaller than 200. Click to see the table. Can you produce some of these chains?
| n | Longest chain |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 4 |
| 6 | 6 |
| 7 | 6 |
| 8 | 7 |
| 9 | 8 |
| 10 | 9 |
| 11 | 9 |
| 12 | 11 |
| 13 | 11 |
| 14 | 12 |
| 15 | 13 |
| 16 | 14 |
| 17 | 14 |
| 18 | 16 |
| 19 | 16 |
| 20 | 17 |
| 21 | 18 |
| 22 | 19 |
| 23 | 19 |
| 24 | 21 |
| 25 | 21 |
| 26 | 22 |
| 27 | 23 |
| 28 | 24 |
| 29 | 24 |
| 30 | 26 |
| 31 | 26 |
| 32 | 27 |
| 33 | 28 |
| 34 | 28 |
| 35 | 29 |
| 36 | 30 |
| 37 | 30 |
| 38 | 30 |
| 39 | 31 |
| 40 | 32 |
| 41 | 32 |
| 42 | 34 |
| 43 | 34 |
| 44 | 36 |
| 45 | 37 |
| 46 | 37 |
| 47 | 37 |
| 48 | 39 |
| 49 | 39 |
| 50 | 41 |
| 51 | 42 |
| 52 | 43 |
| 53 | 43 |
| 54 | 44 |
| 55 | 45 |
| 56 | 46 |
| 57 | 47 |
| 58 | 47 |
| 59 | 47 |
| 60 | 49 |
| 61 | 49 |
| 62 | 49 |
| 63 | 50 |
| 64 | 51 |
| 65 | 51 |
| 66 | 53 |
| 67 | 53 |
| 68 | 54 |
| 69 | 55 |
| 70 | 57 |
| 71 | 57 |
| 72 | 58 |
| 73 | 58 |
| 74 | 58 |
| 75 | 59 |
| 76 | 60 |
| 77 | 61 |
| 78 | 63 |
| 79 | 63 |
| 80 | 64 |
| 81 | 65 |
| 82 | 65 |
| 83 | 65 |
| 84 | 66 |
| 85 | 66 |
| 86 | 66 |
| 87 | 67 |
| 88 | 69 |
| 89 | 69 |
| 90 | 70 |
| 91 | 71 |
| 92 | 72 |
| 93 | 73 |
| 94 | 73 |
| 95 | 73 |
| 96 | 74 |
| 97 | 74 |
| 98 | 75 |
| 99 | 76 |
| 100 | 77 |
| 101 | 77 |
| 102 | 79 |
| 103 | 79 |
| 104 | 81 |
| 105 | 82 |
| 106 | 82 |
| 107 | 82 |
| 108 | 83 |
| 109 | 83 |
| 110 | 85 |
| 111 | 86 |
| 112 | 87 |
| 113 | 87 |
| 114 | 89 |
| 115 | 89 |
| 116 | 90 |
| 117 | 91 |
| 118 | 91 |
| 119 | 92 |
| 120 | 93 |
| 121 | 93 |
| 122 | 93 |
| 123 | 94 |
| 124 | 95 |
| 125 | 95 |
| 126 | 96 |
| 127 | 96 |
| 128 | 97 |
| 129 | 97 |
| 130 | 99 |
| 131 | 99 |
| 132 | 100 |
| 133 | 100 |
| 134 | 100 |
| 135 | 101 |
| 136 | 103 |
| 137 | 103 |
| 138 | 105 |
| 139 | 105 |
| 140 | 106 |
| 141 | 106 |
| 142 | 106 |
| 143 | 106 |
| 144 | 107 |
| 145 | 108 |
| 146 | 108 |
| 147 | 110 |
| 148 | 111 |
| 149 | 111 |
| 150 | 112 |
| 151 | 112 |
| 152 | 113 |
| 153 | 115 |
| 154 | 116 |
| 155 | 116 |
| 156 | 117 |
| 157 | 117 |
| 158 | 117 |
| 159 | 117 |
| 160 | 118 |
| 161 | 119 |
| 162 | 120 |
| 163 | 120 |
| 164 | 121 |
| 165 | 123 |
| 166 | 123 |
| 167 | 123 |
| 168 | 124 |
| 169 | 124 |
| 170 | 126 |
| 171 | 127 |
| 172 | 128 |
| 173 | 128 |
| 174 | 131 |
| 175 | 132 |
| 176 | 133 |
| 177 | 133 |
| 178 | 133 |
| 179 | 133 |
| 180 | 134 |
| 181 | 134 |
| 182 | 135 |
| 183 | 135 |
| 184 | 137 |
| 185 | 137 |
| 186 | 139 |
| 187 | 139 |
| 188 | 140 |
| 189 | 141 |
| 190 | 142 |
| 191 | 142 |
| 192 | 143 |
| 193 | 143 |
| 194 | 143 |
| 195 | 144 |
| 196 | 145 |
| 197 | 145 |
| 198 | 147 |
| 199 | 147 |
| 200 | 148 |
There are many different possible longest chains - at the OEIS you can see the 12 different possible chains, of length 11 (the longest chain - see below), which use numbers from 1 to 13.
Josh then investigated numbers which should not go in chains, and exploited these to develop some winning strategies for playing the game. Josh also created some longest-possible chains using numbers 1 to n, for several values of n. Click here to see Josh's work, and here for a later addition to Josh's research.
Why play this game?
This game can replace standard practice exercises on finding factors and multiples. In order to play strategically, learners think about numbers in terms of their factors, utilising primes and squares to develop winning moves. The switch from a competitive to a collaborative game gives an opportunity for learners to work together and try to find the longest chain, a problem that they could keep coming back to over and over again!
The game also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians. The collaborative version lends itself particularly to fostering a positive attitude to mathematics as learners' resilience may be tested!
Possible approach
The following printable worksheets may be useful: Factors and Multiples Game, Factors and Multiples Puzzle, resources page for 1-100 square grids.
This activity featured in an NRICH video for students in June 2020 and an NRICH student webinar in November 2020. It also featured in an NRICH webinar for teachers in June 2020.
Dozens may be a good problem for students to tackle, before embarking on this challenge.
Play the game as a class, on the board, to introduce the rules, perhaps dedicating the last twenty minutes of each lesson for a week, to playing in pairs. When pupils have finished a game, they could play the next game against someone they've not yet played. At the end of each game, ask pairs to analyse why the last few moves led to its end - working out better moves that could have been made.
To start with you could choose not to mention the initial rule that restricts the starting number to a positive even number that is less than 50. When pupils discover that the first player can win after just three numbers have been crossed, discuss the need to restrict the initial number to an even number smaller than 50.
As learners are playing the game, listen out for those who are considering the probable next few moves when placing a counter/crossing out a number. Game strategies form a natural context for developing deductive logic. You may want to invite a pair of pupils to play against another pair. This gives a 'reason to reason' as each player will need to justify their choice of next move in order for their partner to agree it is the best way to proceed.
The collaborative challenge could run for an extended period: the longest sequence can be displayed on a wall or noticeboard and pupils can be challenged to improve on it. Any improved sequences can be added (perhaps once they have been checked by someone else!). You could then set aside some time in a future lesson to discuss the longest sequence. Does the class think that it is possible to create a longer one? Why or why not?
Key questions
Do you have any winning strategies?
Are there any numbers you should try to avoid?
Possible support
Use a smaller number board, eg 1-50 (or 1-49 in a square). Here is a large 1-50 grid and here is a sheet of smaller grids which could be given to pupils. This makes the mental calculations much easier, without watering down the mathematics. The lesson could focus on accuracy of calculation - with teacher interventions to get pupils sharing their mental strategies.
Handouts for teachers are available here (Word document, pdf), with the problem on one side and the notes on the other.
Possible extension
Xavi's T-shirt is a suitable follow-up problem
Got It
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Problem
Got It is an adding game for two players. You can play against the computer or with a friend. It is a version of a well-known game called Nim.
Start with the Got It target 23.
The first player chooses a whole number from 1 to 4.
Players take turns to add a whole number from 1 to 4 to the running total.
The player who hits the target of 23 wins the game.
Play the game several times.
Can you find a winning strategy?
Can you always win?
Does your strategy depend on whether or not you go first?
To change the game, choose a new Got It target or a new range of numbers to add on.
Test out the strategy you found earlier. Does it need adapting?
Can you work out a winning strategy for any target?
Can you work out a winning strategy for any range of numbers?
Is it best to start the game? Always?
Away from the computer, challenge your friends:
One of you names the target and range and lets the other player start.
Extensions:
Can you play without writing anything down?
Consider playing the game where a player CANNOT add the same number as that used previously by the opponent.
Getting Started
Try to work out the 'stepping stones' that you must 'land on' on your way to the target?
Alter the settings on the game to have a lower target and a shorter range of numbers (e.g. a target of $10$ using the numbers $1$ and $2$). As you play, note down the running totals to refer back to later.
Student Solutions
Students at Clifton Hill Primary in Australia worked hard to beat the computer at Got It with a target of 23, and with numbers 1, 2, 3 and 4 to choose from.
J and C said:
You can win the game with a target number of 23 and range numbers of 1-4 by trying as hard as possible to land on the number 18. By landing on 18, you make your opponent off by 1 of getting the end number. If you get the number 18, your opponent will be stuck and you have a guaranteed win.
Danica and Eve took this a little bit further:
Can you find a winning strategy?
Yes, you can by getting to 13 then the other player has to say their number (1-4) you can get to 18.
After that, no matter what number they say there will always be a way for you to get to 23.
Leon and Jack built on this even further:
There are three key numbers:
8
13
18
Firstly, you have to be the one that gets the total to eight. You can do this by starting with 3 and the opponent has to let you get to 8. Once you have accomplished that, it is their turn with a total of 8. They then can't stop you from getting to 13. After you get 13 they can't stop you from getting to 18. After this, they can't stop you from reaching 23.
So, in fact rather than there being three ‘key numbers’, it sounds like you’re saying there are four key numbers: 3, 8, 13, 18.
Golden Eagles Class at Ditchingham Primary Academy described their strategy in a similar way:
We found that it was easier to win once we had found the stepping stones for the target number. This was a key number that if we reached we could win any game.
As we could add up to 4 each time the difference between each number was 5. The stepping stones were 3, 8, 13, 18 for 23.
For example, if we started with 3, our opponent could only make it up to 7, we could then add 1 to make 8. The most they could then make is 12 but with all variation of numbers we could then make 13 and so on up to 23.
When we played the computer using this strategy and we went first we always won, however if the computer started we always lost. It was possible to beat an opponent if they went first, but only if they hadn't worked the stepping stones out. Once they had we always drew.
We tried using 17 as a target number and found stepping stones at 2, 7, 12 using the numbers 1-4. This was again as the difference between these numbers was 5.
We would like to try dropping the 4, using numbers 1-3 and seeing if the stepping stones reduced to a difference of 4.
I like the fact that you are making a conjecture about how you could apply your strategy to a version of the game using only numbers 1, 2 and 3, Golden Eagles Class. This is exactly what mathematicians do!
This idea of ‘key numbers’ of ‘stepping stones’ is crucial in this game. Mateusz and Joseph from St Ann’s Primary agreed with this approach, so did Amelie and Sam from St Francis of Assisi Primary School; Gabriel from Yavneh College; Alaric from Haydon Wick Primary; Harry from St Michael’s Preparatory School; Ruby, Alice, Meiyla, Lilly, Olivia and Sophie from Anston Greenland’s Primary; Joy, Akosua, Asuo and Kumi from Torrens Valley Christian School in Australia; as well as three groups of students from Ganit Kreeda, Vicharvatika, India.
Anirudh from Hamilton Primary School shared this video, in which he explains how he worked backwards to find the stepping stones:
And in this video, Kaana from Ganit Kreeda explains the importance of multiples of 5:
Can you see how Anirudh’s and Kaana’s solutions are connected?
Joseph from Jesmond Park Academy sent a very comprehensive solution:
I initially approached this problem by just playing some games with the computer and tried to recognise some patterns. I started by playing first but then swapped to the computer playing first. It was here that I noticed that it would always start with the same first number depending on the target number.
I noted this down and also noted that it would try to reach certain values consistently. I called these 'landing values' and wrote them down.
From here, I noticed that there was a fixed distance between each landing value that was 1 greater than the largest number I could choose. This revelation led to further investigations that led to the process below.
For anyone who wanted to consider solving this problem, or ones like it, I think the best first step is to just play around and consider different values. After you've considered different values, you can assess whether there is a pattern or not. This pattern is the real starting off point for finding a general solution.
Thank you, Joseph, what a lovely description of the process you went through. We agree that ‘playing around’ with a new problem is a great way to start. The exploration then leads to some noticing, as Joseph said, and then some more systematic working, some generalising and even proof.
Here is Joseph’s general strategy for winning at Got It:
The numbers that are chosen to be added up range from 1 -> n, where n is a positive integer and the spaces between 1 and "n" are integers. (In fact I wonder whether they have to be consecutive numbers, Joseph...)
For example, "1, 2, 3, 4" where n is 4.
The player will have a target number, which we will call "m". To begin, player 1 will write down the target and subtract (n+1) away from the number "m" until there is a number that they are able to choose, we will call this "b".
For example, consider the target is 22 (so m=22) and the player can add up to 4 (so n=4), they will subtract 5 (n+1) until they reach the number 2 (so b=2).
Player 1 will choose this number and begin the game. From this point forward, it is Player 1's objective to reach the next multiple of (n+1) that starts from "b". In our example, these "Landing Values" would be 7, 12, 17 and finally, 22. Player 1 will always be able to force a Landing Value on their turn assuming they started from "b" and their opponent plays perfectly.
If player 1 aims for these numbers, Player 2's choice will not matter as they will end up unable to reach the target number.
The exception to this trend comes if the target number is a multiple of (n+1). This is because you can't keep subtracting (n+1) to reach the initial number "b" as you will eventually reach (n+1) and the next number would be 0.
For example, if we could choose to add up numbers between 1 and 4, if the target was 10, we could subtract 5 (n+1) once.
Following this, subtracting 5 again would reach 0, which isn't an option with our initial conditions.
Thus, the first player would have to choose any number between 1 and 4.
Assuming that there is a perfect opponent, they would reach the multiple of (n+1) instead, in this example it would be 5. The game would continue as stated previously, resulting in player 2 winning instead.
Thus, you can reach the conclusion that it is only sensible to be Player 1 if the target number is not a multiple of (n+1). If it is, then you should choose to be Player 2.
Abishai expressed a general solution in a very similar way, so did Xavier, and so did students from Ganit Kreeda, India. Thank you to you all. How fantastic that we have used our mathematics to find a winning strategy that will ALWAYS work when playing Got It.
Teachers' Resources
Why play this game?
Got It is a motivating context in which learners can apply simple addition and subtraction. However, the real challenge here is to find a winning strategy that always works, and this involves working systematically, conjecturing, refining ideas, generalising, and using knowledge of factors and multiples.
Possible approach
This problem featured in an NRICH Secondary webinar in June 2021.
Introduce the game to the class by inviting a volunteer to play against the computer. Do this a couple of times, giving them the option of going first or second each time (you can use the "Change settings" button to do this).
Ask the students to play the game in pairs, either at computers or on paper. Challenge them to find a strategy for beating the computer. As they play, circulate around the classroom and ask them what they think is important so far. Some might suggest that in order to win, they must be on 18. Others may have thought further back and have ideas about how they can make sure they get to 18, and therefore 23.
After a suitable length of time bring the whole class together and invite one pair to demonstrate their strategy, explaining their decisions as they go along. Use other ideas to refine the strategy.
Demonstrate how you can vary the game by choosing different targets and different ranges of numbers. Ask the students to play the game in pairs, either at computers or on paper, using settings of their own choice. Challenge them to find a winning strategy that will ensure they will always win, whatever the setting.
Key questions
Possible extension
Possible support
Square It
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Problem
This game is part of a set of three. We recommend you play this version before having a go at Parallelogram It and then Rhombus It.
Square It printable sheet
Printable dotted grid
This game can be played against a friend or against the computer.
Players take it in turns to click on a dot on the grid - the first player will place blue triangles and the second player will place pink squares.
The winner is the first to have chosen four dots that can be joined to form a square.
Squares can be anywhere and any size.
Clicking on the purple settings cog allows you to display the dots on a coordinate grid, to select the size of the grid, who the players are, and who goes first.
Once you've played a few times against a friend, you might like to discuss your strategies, and then test them by playing against the computer.
Can you find a winning strategy?
If you are not using the interactive game, you may like to print off some dotty paper.
Getting Started
Play with this interactivity until you can draw tilted squares confidently.
You could fix one side of a square and then work out where the other corners need to go.
Student Solutions
Well done to everyone who developed a winning strategy. Olivia from McCauley Catholic High School in England dscribed an idea which most strategies are built on:
All I did was set up another square that I could go to if my first one didn't work.
Rishik had another general tip:
If given a choice a player should try to begin first as it not only increases the success rate by opening many possibilities but does not let the player became obstructed by disruptions. The first player should choose to place their dot on in the centre of their grid as this allows many chances for the player to win.
Tushar from NPS Rajajinagar in India, Finlay form Colyton Grammar School in England, Sydney from Acera, Aaron from the UK and Marco from International School of Lausanne in Switzerland, Nathan and Gabriel from The British School Al Kubairat in UAE and Rishik all described the same strategy. This is Tushar's work:
This is one of the winning strategies that I use. The main goal of this strategy is to get such a position that there are 2 possible squares that you can make. The other player can only stop you from making one square. After the opponent makes their move, you simply connect the other one. This strategy shows the quickest way to get that position. This strategy works if you are player 1 (triangle). The position that you are trying to get is this:
If you get this kind of position, you win. It doesn’t always have to be pointed upwards, it can be in any direction.
Rishik explained how to get to this point, and some other students from International School of Lausanne in Switzerland decided that the game was won before they had finished making the full shape. Vince wrote:
The blue triangles have already won at this point because the red squares see that the triangles are almost done with their square so red tries to block blue. Unfortunately for red it was too late because there are two little tricks that blue can do.
Trick 1
The first and most popular trick is to put a triangle in the middle of the three other triangles. This creates something where blue has two options to win. And when red blocks one of them, blue can use the other one to win.
Trick 2
The second but my personal favourite trick is to add a triangle in the close top right corner to make the same thing as before but just in a different format. In this case, the red player will have to pick a way to lose.
Can you see how closely related these tricks are? Charles, Mark and Yihang from Colyton Grammar and Issa M and William from The British School Al Kubairat described the technique in trick 2. Issa wrote:
First you must place a triangle in the middle of the grid. Then you must make a diagonal line starting from the top right to the bottom left of the middle triangle. Then place another, one space from the bottom left of the middle triangle. This will open up possibilities to make a diamond while the computer will still think that you will make a normal square and will block that.
Lottie from Colyton Grammar School explained how tricks 1 and 2 are the same by describing the strategy that links them:
Place three squares, one on top of the other, to make a straight line. This can be horizantally, diagonally or vertically. Then place another symbol adjacent to the square in the middle of the existing line, this can be on the left or right, it does not affect the method. If your opponent tries to block you by adding a square adjacent to the top square of the line, on the same side that you added a square on to the middle, you can still win in your next go as you have two possible squares you could make. You could place a square next to the bottom square, again on the same side that the other squares have been places. This is the same vise versa. This particular method ensures that you will make a square before your opponent and win.
Rishik explained that there are even more ways to create this shape - you don't actually need touching points - and also described another winning shape:
The player should always try to form two formations: the arrow and parallelogram.
Arrow:
Parallelogram:
Although I have chosen to demonstrate these 2 formations with 3 by 2 grids, they can be formed with any size, by scaling these formations up or rotating or reflecting them in any way on the grid. There are many ways, but these two formations are the best way to secure victory as these create 2 possibilities for you to win and your opponent can only place one counter, thus letting you make a square by completing the other.
Can you see why you have won if you create a "parallelogram"?
Dylan from Chesterton Community College in the UK had a strategy which involved responding to the computer, or opponent:
The method I found does not work on a 3 by 3 grid because the grid is too small. It works on any larger grid. To start you put a dot in the centre of an odd sized grid, or on one of the four centre points of an even sized grid. Then where ever the computer places its dot around your dot, your next dot should be adjacent to that dot on a horizontal or vertical line from your 1st dot. Next you place one above the computer’s next dot. Your 4th dot should be on a diagonal line with your 3rd dot and a horizontal or vertical line with your first dot. This will give you a winning position as you can make two different squares with your next dot.
Lewis from Colyton Grammar School and Millie from Thomas Gainsborough School in the UK used yet another strategy. This is Millie's work:
Teachers' Resources
Why play this game?
This game offers an excellent opportunity to practise visualising squares and angles on grids, and encourages students to develop winning strategies for beating an opponent. Describing strategies to others is always a good way to focus and clarify mathematical thought.
Working with tilted squares provides an opportunity to examine the properties of gradients of parallel and perpendicular lines. This can lead on to Square Coordinates and Opposite Vertices, and Tilted Squares for students who are going on to work on Pythagoras' Theorem.
Possible approach
This game featured in the NRICH Primary and Secondary webinar in April 2023.
Start with a demonstration playing against a student rather than against the computer. Students are often surprised when the winning square isn't aligned with the grid. This leads to discussions about what makes a square a square.
After a demonstration of the game, students could be left to play for a while in pairs, ideally on tablets or computers using the interactivity. If this is not possible, students could use this paper grid.
The Settings menu (purple cog) offers the chance to have different sized grids, and coordinate axes if you prefer.
Bring the class together for a discussion of their thoughts on the game. Did anyone consistently win or lose? Can anyone think of any good strategies which might help them win? Are they able to ensure a win by setting up a situation in which they can create two different squares on the next turn?
Once ideas have been shared the group can return to playing in pairs, ideally against the computer. Encourage each student to explain the reasoning behind the moves they suggest.
If they are using the interactivity, students might like to use the 'Game report' to help them look back on the game and analyse possible alternative moves.
One aspect of developing a winning strategy that could be considered is the number of distinctly different starting points ($6$ on a $5 \times 5$ board) and the number of different squares that can be drawn that include each of those points. This can help students decide on a good place to start.
The final plenary might involve the whole class playing against the computer, putting into practice the strategies that have been discussed.
Key questions
- Why did you make that move?
- Why do you think the computer made that move?
- How do you know a square is a square?
- How could you set up a situation in which you can create two different squares on your next turn?
- Can you beat the computer if the computer goes first?
Possible support
The game can be built up gradually from a 25 dot board and a 36 dot board to the 49 dot board in the question. Students could be asked to draw examples of all the different possible squares on their specific board size, and to compare notes to check for wrong or omitted solutions.
Some students might find 'believing' in the tilted squares difficult. On paper they could use the corner of a piece of paper or a set square, for example, to convince themselves that the angles in a shape are $90^\circ$. Alternatively, they could be encouraged to cut the shapes out and move them around to see if the cut-out really looks square.
Possible extension
Suitable follow up problems involving generalising about squares are Square Coordinates and Tilted Squares. Alternatively, students could move on to other quadrilaterals by working on Parallelogram It, Rhombus It and Opposite Vertices.
