/keybase/public/procspero/[Math] Probability and Statistics/Combinatorial Game Theory/On Numbers and Games, John Horton phisrebiberkotch.gq 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. ^ 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.
It will also discuss how to spot good number games and learn to identify the learning opportunities the games can offer for students.
The first activity is an example of a game that helps to develop the understanding of number relationships. Many such games are freely available in books and on the internet. Before attempting to use the activities in this unit with your students, it would be a good idea to complete all, or at least part, of the activities yourself.
It would be even better if you could try them out with a colleague as that will help you when you reflect on the experience. Activity 1: What do you need? For this activity, students work in pairs or small groups.
Give students a printout or copy of a hundred square Figure 1 , or make them use one they already have in their books. Make sure they can all see the hundred square. It is also important not to tell them how to go about doing this task but to let them discover it for themselves so they have to think about strategies. This also keeps up the suspense. Figure 1 A hundred square Part 1: Deciding on what you need to know Write the following on the blackboard: Eight clues The number is greater than 9.
The number is not a multiple of The number is a multiple of 7. The number is odd. The number is less than Its ones digit is larger than its tens digit.
Its tens digit is odd.
Tell the students the following: I have got a number in my mind that is on the hundred square but I am not going to tell you what it is. However, you can ask me four of the eight clues that are written on the blackboard and I will answer with yes or no. There is something else I need to tell you: four of the given clues are true but do nothing to help in finding the number.
Four of the clues are necessary for finding it. Can you find out, in your group, the four clues that help and the four clues that do not help in finding the number I am thinking of? What is it that you need to know to find a chosen number from your hundred square? Part 2: What is the number? Whether they work or not, ask for the reason they chose those four clues.
Ask if a group has a different set of clues and try those out. Stop when the clues work and discover your chosen number.
Repeat this for several different numbers, or ask a student to take over your role and think of a number. Follow this with a discussion of which four clues are not needed to find the number, and why this is. What makes the best clues, and why?
Video: Using local resources Case Study 1: Mrs Bhatia reflects on using Activity 1 This is the account of a teacher who tried Activity 1 with her elementary students I do agree that games are fun activities to do but I am always a bit sceptical when it is claimed they can offer rich opportunities for learning mathematics.
I found it hard to believe that these number games could supplement the learning that happens in the normal traditional teaching I do in my classroom, when I explain mathematical ideas clearly to my students. But I decided to give it a go because I do find that even my younger students seem bored at times when doing mathematics, and that saddens me and makes me feel I have to try something else.
My class is large — about 80 students. Although they are supposed to be only from Classes III and IV, the attainment between the students varies a lot: some of the students seem to still struggle with early-year concepts of numbers, others are happy and able to do the work of higher classes.
Finding activities that all of them can do and that challenge them all is very hard. As I do not have access to resources such as a photocopier or large sheets of paper or scales for all students, in preparation for this activity I asked the students to draw a hundred square on the inside cover of their exercise books for homework.
Most of the students had already done this, but some had not. I did not want to spend any time in the lesson letting those students draw one there and then so I made sure that when the students went into groups of four or five they were sat right next to someone who had done the homework so they could both look at the same hundred square.
To move them into groups I simply asked the odd rows of students to turn around and work with the students sitting opposite them. I was very uncomfortable with not giving the students precise instructions on how to go about finding out which clues were necessary or not, or to discuss this with the whole class first.
I was really worried they would not know what to do. But I thought I would try and see if the activity would work and presented the task in the way it was described. I decided that if the students still did not know how to proceed after four minutes, I would tell them.
TI-AIE: Using number games: developing number sense
It did not take them that long. To make sure they were on-task I walked through the classroom and listened in on their discussions. I noticed that neighbouring groups would listen to their replies as well, sometimes changing their approaches. 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.
J.H. Conway, R.K. Guy - The Book of Numbers
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. It has been extended to include a survey of results on misere games, a list of open problems involving them, and a summary of MisereSolver [AS], the excellent Java-language program for misere indistinguishability quotient construction recently developed by Aaron Siegel.
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.
What this unit is about
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.
Software There is considerable mystery surrounding the category of commutative monoids that arise as misere quotients of impartial combinatorial games.
We're quite interested in finding a structural theory.Although they are supposed to be only from Classes III and IV, the attainment between the students varies a lot: some of the students seem to still struggle with early-year concepts of numbers, others are happy and able to do the work of higher classes. Finding activities that all of them can do and that challenge them all is very hard.
This page was last edited on 13 March , at Because we do not have dice in the school, I made the spinners myself. 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.
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