- Contents
- On Numbers and Games
- John H. Conway
- Surreal Numbers and Games
- On Numbers and Games (pdf) | Paperity

/keybase/public/procspero/[Math] Probability and Statistics/Combinatorial Game Theory/On Numbers and Games, John Horton prehexfejefne.ga On Numbers and Games by John H. Conway. Read online, or download in secure PDF format. are the wonderful books On Numbers and Games [ONAG] by Conway, and Winning Keywords: Conway games, surreal numbers, combinatorial game theory.

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On Numbers and Games. ByJohn H. Conway. Edition 1st Edition to read online and download this title. DownloadPDF MB Read online. On Numbers and Games is a mathematics book by John Horton Conway first published in References[edit]. ^ Fraenkel, Aviezri S. (). "Review: On numbers and games, by J. H. Conway; and Surreal numbers, by D. E. Knuth" ( PDF). Portrait on Numbers and Games by Conway - Download as PDF File .pdf) or read online.

Toggle navigation. New to eBooks. How many copies would you like to download? On Numbers and Games by John H. Add to Cart Add to Cart. Add to Wishlist Add to Wishlist. ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century.

Many wild misere games that have long appeared intractible may now lie within the grasp of assiduous losers and their faithful computer assistants, particularly those researchers and computers equipped with MisereSolver. Along the way, we illustrate how to use the theory to describe complete analyses of two wild taking and breaking games.

The previous papers all treat impartial misere games. For a canonical theory of partizan misere games, start here: Mar Misere canonical forms of partizan games [Aaron Siegel] Abstract: We show that partizan games admit canonical forms in misere play.

The proof is a synthesis of the canonical form theorems for normal-play partizan games and misere-play impartial games. It is fully constructive, and algorithms readily emerge for comparing misere games and calculating their canonical forms.

We use these techniques to show that there are precisely games born by day 2, and to obtain a bound on the number of games born by day 3. In that way they all learned from each other without having to stop the whole class to discuss this. I really liked the buzz in the class — there was excitement and engagement. The students were smiling a lot and developing their mathematical arguments to discuss, agree and disagree with each other.

Everyone seemed to have a point to make and seemed to be included in the work of the groups. After a while I told them they had three more minutes to decide on the clues, and that they had to make sure that everyone in their group knew and understood what the agreed clues were.

I asked that because I wanted to make sure that all students in the group, whatever their attainment, would learn from this game. The discussion about why the clues were needed or not needed gave the students the opportunity to talk about their ideas. If they did not agree, they had to say why.

Reflecting on your teaching practice When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and being able to progress, and those you needed to clarify. If they do not understand and cannot do something, they are less likely to become involved.

Do this reflective exercise every time you undertake the activities, noting, as Mrs Bhatia did some of the smaller things that made a difference. Pause for thought Good questions to trigger reflection are: How did it go with your class?

Which questions were most successful in enabling students to demonstrate their mathematical thinking? Did all your students participate? Did you feel you had to intervene at any point?

Did you modify the task in any way? If so, what was your reasoning for doing so? To help in deciding which games offer good mathematical learning for use in the classroom it is helpful to first think about the characteristics of good educational games in general. Activity 2 presents some games that help to develop an understanding of number relationships. Many such games can be found freely in books and on the internet. Activity 2: Being strategic about numbers Preparation This game asks the students to think about place value and is enjoyed by students of all ages.

For younger students the size of the boxes can be reduced. Several variations to the game set-up and scoring systems are suggested. Once the students understand the set-up you can also ask them to come up with more variations and scoring systems of their own, as these will also require mathematical thinking. For this activity students will need six-, nine- or ten-sided dice with numbers 1 to 6, 1 to 9 or 1 to 10 or spinners with ten segments numbered 1 to 10 or 0.

You can find templates for spinners in Resource 3 These resources can be used again in Activity 4. Game 1 below describes how to set up the basic game, and Games 2 to 6 describe variations and developments from Game 1. Playing the games This game is best played in pairs, or with two pairs playing against each other. Each player draws a set of four boxes, as shown in Figure 2.

Figure 2 Each player has a set of four boxes.

Instruct the students as follows: Take turns to roll the dice, read the number and decide which of your four boxes to fill with that number. 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. Here are two possible scoring systems: One point for a win. The first person to reach 10 points wins the game. Work out the difference between the two four-digit numbers after each round. The winner keeps this score. First to 10, wins. Game 2 Whoever makes the smaller four digit number wins. Game 3 Set a target to aim for.

Then the students throw the dice four times each and work out how far each of them is from the target number. Whoever is the closer to the target number wins. Work out the difference between the two four-digit numbers and the target number after each round. Keep a running total. First to 10, 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 number. The winner is the one closest 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, different scoring systems are possible. Game 5 This game really requires strategic thinking and can be very competitive!

Tell your students the following: 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. This variation of the game becomes even more challenging when you play it with more than two people.

Game 6 This is a cooperative game rather than a competitive one — to be played by three or more people. Tell your students the following: 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. I discussed this with a colleague and we decided to first try it out ourselves in the staff room.

And oh my, is it fun to play! We could hardly stop, and other teachers had a go as well. I was a little bit worried about making up teams of younger and older students as I teach mixed-age groups, so when we first played the game I made students of similar age play in pairs against each other. We played game 1 and then game 2, each one a couple of times. Since then we have used these and other games regularly, sometimes at the beginning of the lesson to energise the students especially good after lunch , and sometimes at the end of the lesson.

I initially thought it would help the older ones in their learning because they would have to help and communicate their mathematical thoughts with the younger students, and that has indeed been the case.

At the same time I realised I made the assumption that the younger ones would be reluctant to talk with the older students — but that has proved wrong! The younger students are very happy arguing with the older ones about the mathematics involved.

Because we do not have dice in the school, I made the spinners myself. I made them on cardboard and they have now been used often, so it was worth the effort. Work out the difference between the two four-digit numbers after each round. The winner keeps this score. First to 10, wins. Game 2 Whoever makes the smaller four digit number wins. Game 3 Set a target to aim for. Then the students throw the dice four times each and work out how far each of them is from the target number.

Whoever is the closer to the target number wins. Work out the difference between the two four-digit numbers and the target number after each round. Keep a running total. First to 10, 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 number. The winner is the one closest 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, different scoring systems are possible.

Game 5 This game really requires strategic thinking and can be very competitive! Tell your students the following: 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. This variation of the game becomes even more challenging when you play it with more than two people. Game 6 This is a cooperative game rather than a competitive one — to be played by three or more people. Tell your students the following: 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.

I discussed this with a colleague and we decided to first try it out ourselves in the staff room. And oh my, is it fun to play! We could hardly stop, and other teachers had a go as well. I was a little bit worried about making up teams of younger and older students as I teach mixed-age groups, so when we first played the game I made students of similar age play in pairs against each other. We played game 1 and then game 2, each one a couple of times.

Since then we have used these and other games regularly, sometimes at the beginning of the lesson to energise the students especially good after lunch , and sometimes at the end of the lesson. I initially thought it would help the older ones in their learning because they would have to help and communicate their mathematical thoughts with the younger students, and that has indeed been the case.

At the same time I realised I made the assumption that the younger ones would be reluctant to talk with the older students — but that has proved wrong!

The younger students are very happy arguing with the older ones about the mathematics involved. Because we do not have dice in the school, I made the spinners myself.

I made them on cardboard and they have now been used often, so it was worth the effort.

I would like to make one big dice that I can roll and then all the students would have to work with the same numbers — just as a variation on the game. I had never really thought about it in detail.

This strategic thinking really helped them to develop their understanding of place value because they had to think very carefully about the value of each digit. Pause for thought In the case study, Mr Mehta was positive about the interaction between the older and younger students in his class.

Reflect about how your own lesson s went using some of these questions: What did you like about these activities? What is it about these tasks that make students want to participate and engage? What mathematical learning opportunities did these activities offer? Is there anything you would like to add or modify? Make some notes of your thoughts and ideas in response to these questions and discuss them with the teachers in your school or at a cluster meeting.

The games used so far in this unit have offered mathematical learning opportunities — that is, the games helped the students to develop their understanding of specific mathematical concepts and ideas — in this case, number sense. This means that games are not only fun for the students but are also a valid way to learn mathematics.

For example, in Activity 2 the mathematical learning opportunities for the students can be described as: learning about place value learning about the magnitude of numbers learning to use mathematical operations efficiently and accurately learning to work with numbers flexibly and fluidly.

Learning about these mathematical ideas is of considerable importance in the curriculum and essential for developing number sense. The next activity builds on the game in Activity 2.

The learning opportunities are extended to working with different operations, understanding number relationships and the effect of different mathematical operations on numbers. Again there are several variations to choose from. These games are best played in pairs, or with two pairs playing against each other. For this activity students will again need six-, nine- or ten-sided dice with numbers 1 to 6, 1 to 9 or 1 to 10 , or spinners with ten segments numbered 1 to 10 or 0.

You can find templates for spinners in Resource 3. Instructions for all games Students take turns to throw the dice or turn the spinner and decide which of their cells on the grids to fill in. This can be done in one of two ways: either fill in each cell as you throw the dice, or collect all your numbers and then decide where to place them. Playing the games Each of the students draws an addition grid like Figure 3. Figure 3 An addition grid.

Throw the dice nine times each until all the cells are full. Whoever has the sum closest to 1, wins. There are two possible scoring systems: One point for a win. First to 5, loses. You can vary the target to make it easier or more difficult, or you can get the class to practise using negative numbers above 1, positive, below 1, negative and suggest that the team closest to zero after ten rounds wins.

Game 2 Each of the students draws a subtraction grid like Figure 4. Figure 4 A subtraction grid. Throw the dice eight times each until all the cells are full. Whoever has the difference closest to 1, wins.

You can vary the target to make it easier or more difficult, perhaps including negative numbers as your target. Game 3 Each of the students draws a multiplication grid like Figure 5. Figure 5 A multiplication grid.

Throw the dice four times each until all the cells are full.