Mathematical Games A Rationale for their Use in the Teaching of Mathematics in School What would be the advantages? •Motivation. Games generate enthusiasm, excitement, total involvement and enjoyment and, over a period of time, should enhance pupils’ attitudes towards the subject. •Variety. Games add variety to the overall mathematics curriculum, by bringing another varied approach into the teaching of the subject. •Discussion. Games encourage discussion. •Co-operation. Even competitive games can encourage cooperation. •Active involvement. Games encourage the active involvement of children, making them more receptive to learning and increasing their motivation. Active involvement not only enhances learning, but according to some psychologists is essential for learning to take place at all. For this reason psychologists including Piaget, Bruner and Dienes suggest that games have a very important part to play in learning, particularly in the learning of mathematics. Of these three, Zoltan P. Dienes goes furthest by suggesting that all mathematics teaching should begin with games. Although Dienes may be overstating his case, he is a man well worth listening to. Dienes Dienes has not only carried out an extensive programme of classroom research, he has also developed some of the best apparatus available for teaching mathematics, including the multi-base arithmetic blocks, the algebraical experience materials, logic blocks and the number balance. I have claimed that, if we can teach mathematics through games, then there are many desirable by-products. But can mathematics be taught effectively using games? Leaving aside general aims such as those above, the major purpose of teaching mathematics is the attainment of objectives. Let us focus on three type of objective. 1. The Reinforcement and Practice of Skills Much of mathematics teaching revolves around giving children practice in newly acquired skills, or in reinforcing and further developing skills. Games provide a way of taking the drudgery out of the practice of skills, and indeed of making the practice more effective. (See Steeplechase on the handout.) 2. The Acquisition and Development of Concepts See Fair/Unfair Games on the handout. See Edith Biggs’ research project as discussed in the handout. See Steeplechase on the handout. The sample studies discussed in the handout show how games can play a vital part in aiding children to first acquire and then to further develop mathematical concepts. 3. The Development of Problem Solving Strategies HMI have specified the following problem solving strategies as distinct objectives of mathematics teaching: •Trial and error methods •Simplifying difficult tasks •Looking for pattern •Making and testing hypotheses •Reasoning •Proving and disproving Mathematical games can foster the development of most, if not all, of these strategies and higher level skills. Analysing Games Level 1: Local reasoning Each time we make a move we have to ask ourselves what the immediate consequences of that move are likely to be: “If I go there, then he/she will …”. This kind of reasoning is local in the sense that we apply it to one little bit of the whole game at a time. Such reasoning is important, but it ignores long-term effects. A move may be locally safe, yet guarantee defeat in the long run! Level 2: The search for global rules Global rules or strategies are those which influence one’s playing of the game as a whole. Level 3: Being absolutely sure Here we need some kind of mathematical proof that one’s strategy really does control play in the way one thinks it does. THE “FIRST TO 100” GAME This is a game for two players. Players take turns to choose any whole number from 1 to 10. They keep a running total of all the chosen numbers. The first player to make this total reach exactly 100 wins. Sample Game: Player 1’s choice Player 2’s choice 10 10 5 8 15 23 8 2 31 33 9 9 42 51 9 8 60 68 9 9 77 86 10 4 Running Total 96 100 So Player 1 wins! Play the game a few times with your neighbour. (Play & modify) Can you find a winning strategy? Try to modify the game in some way, e.g. - suppose the first to 100 loses and overshooting is not allowed. - suppose you can only choose a number between 5 and 10. The Spiral Game This is a game for two players. Place a counter on the dot marked “Start”. Now take it in turns to move the counter between 1 and 6 dots inwards along the spiral. The first player to reach the dot marked “Finish” wins. Try to find a winning strategy. Change in some way the rule for moving, and investigate winning strategies. Start Finish First One Home This game is for two players. You will need to draw a large grid like the one shown. Place a counter on any square of your grid. Now take it in turns to slide the counter any number of squares due West, South, or SouthWest. The first player to reach the square marked “End” is the winner. End Pin Them Down! A game for 2 players. Each player places his/her counters as shown. The players take it in turns to slide one of their counters up or down the board any number of spaces. No jumping is allowed. The aim is to prevent your opponent from being able to move by trapping his/her counters. Domino Square This is a game for 2 players. You will need a supply of 8 dominoes or 8 paper rectangles. Each player, in turn, places a domino on the square grid, so that it covers two horizontally or vertically adjacent squares. After a domino has been placed, it cannot be moved. The last player to be able to place a domino on the grid wins. NIM This is a game for 2 players. Arrange a pile of counters arbitrarily into 2 heaps. Each player in turn can remove as many counters as (s)he likes from one of the heaps. (S)he, can if (s)he wishes, remove all the counters in a heap, but (s)he must take at least one. The winner is the player who takes the last counter. Try to find a winning strategy. Now change the game in some way and analyse your own version. Laser Wars Two tanks are armed with laser beams that annihilate anything which lies to the North, South, East or West of them. They move alternately. At each move a tank can move any number of squares North, South, East or West but it cannot move across or into the path of the opponent's laser beam. A player loses when he is unable to move on his turn. Play the game on the board shown, using two objects to represent “tanks”. Try to find a winning strategy which works wherever the tanks are placed to start with. Try to change the game in some way. Kayles This is like an old 14th century game for 2 players, in which a ball is thrown at a number of wooden pins standing side by side. The size of the ball is such that it can knock down either a single pin or two pins standing next to teach other. Players alternately roll a ball and the person who knocks over the last pin (or pair of pins) wins. Try to find a winning strategy. (Assume that you can always hit the pin or pins that you aim for, and that no one is ever allowed to miss). Now try changing the rules (Alternative Presentation) Kaylox Draw out a connected line of cells, such as: Decide who is to be O and who is to be X. Players take turns. On each turn, a player must put his/her mark in either 1 square or 2 adjacent squares. No square may be used twice. The player who makes the last mark, or marks, is the winner. Towers of Hanoi A puzzle for one person. In a temple at Benares there were three rods and one rod held 64 discs of gold, all of different diameters, placed so that the largest lay at the bottom and the others, in decreasing order of size, rested upon it. The priests were set the task of moving the discs, one at a time, so that eventually the discs would rest in the same order on the other rod. At no one time could a disc be placed upon a smaller one. About how long do you think the task would take them, assuming that they were to work without stopping and that the time taken to move a disc from one rod to another was five seconds on average? What is the least number of moves necessary to move two, three, four, ..., sixty-four, ..., n, ... discs from one rod to another? Can you prove the result? Sprouts This is a game for two players. All that is needed is a plain piece of paper and a pencil. To start, mark a number of dots on the paper; it is best to begin with three dots, but try any number from 2 to 8. Each player takes it in turn to draw a line which joins one dot to any other dot, or to itself, and then places a new dot anywhere on this line. These restrictions must be observed: (a) The line must not cross itself or any other line, nor pass through any other dot; (b) No dot may have more than three lines coming from it. The winner is the last person able to play. Is there a rule which determines the number of moves which can be made in any game? MISOX Draw a 3x3 grid as used for Noughts and Crosses. Decide who is to be O and who is to be X. Players take turns putting their own marks in, only one mark at a time. The player who first gets three of his marks in a straight line, vertically, horizontally or diagonally, loses the game. QUOX Draw out a grid of 3 x 3 squares. Decide who is to be O and who is to be X. Players take turns to put their marks in as many squares as they like provided that the squares used are all in the same straight line (vertically or horizontally). They do not have to be next to each other. No square may be used twice. The player who makes the last mark, or marks, is the winner. X X O O X O RINGOX Draw out a connected “chain” of cells. The actual number is not important. Decide who is to be O and who is to be X. Players take turns to put their mark in 1, 2 or 3 adjacent cells. Each cell may have only 1 mark. The player who marks in the last cell, or cells, is the winner. X X O X O X O O O O X X Cat & Mouse A game for two players. The board is made up of 27 ‘holes’ connected by ‘passages’. Each player has a counter or marker of their own. One player is the mouse, the other player is the cat. At the beginning the cat goes on C and the mouse goes on M. Players then take turns moving their own markers. Cat goes first. Moves are made from one hole to the next along the passages. The cat captures the mouse if it can move into the same hole as the mouse. The mouse tries to avoid being caught! A good mouse is never caught! Spiralin’ A game for two players. Place 4 counters or markers on the spots shown circled. Players take turns moving these markers. In their turn, a player must move one marker by sliding it along the line (towards the centre) a distance of 1, 2 or 3 spots. Markers may not jump or overtake, and no spot may have more than 1 marker on it. A player moving a marker on to the centre spot takes off that marker. The winner is the player who takes off the last marker. Square Dance A game for two players. Players take opposite corners and place 2 of their own counters or markers on the dots which are circled. One marker on each dot. Players take turns moving one of their own markers at each turn. A marker may be moved 1, 2 or 3 dots forwards or backwards around the square. Two markers cannot be put on the same dot, and they cannot jump or pass each other. A player who is unable to make a move loses the game. A game for two players. Place a counter or marker on every square except the one with the star. Players take turns. At each turn a player must move one marker by jumping over one other marker into an empty square. This move may be up, down, or across, but not diagonally. The marker that has been jumped over is removed. The last player who is able to make a jump wins the game. Take One! Take Two! A game for two players. Place a counter or marker in each of the 25 circles. Players take turns removing exactly one pair of markers at each turn. The pair removed must be in two circles that touch. The player who removes the last pair wins the game. Accumulator One counter is needed. The first player starts by placing the counter on one of the numbers and saying that number. Then, starting with the second player, each player in turn moves the counter by sliding it along a straight line to another number and saying the total so far. When a total of (say) 23 is reached, that player wins. If the total exceeds 23, the player loses. Star Pick Place a counter on each spot. Players take turns picking up these counters. In his turn a player may pick up 1 or 2 counters. A player may only pick up 2 counters provided that they are connected by a single straight line. The winner is the player who picks up the last counter(s). Poly Pick Place a counter on each spot. One player is White, the other is Black. Players take turns picking up the counters. In his turn a player may pick up 1 or 2 counters. A player may only pick up 2 counters provided that they are connected by a single straight line of the player’s own colour. The winner is the player who picks up the last counter(s). ‘Modern’ Seega This is a game played by young Egyptians today. Two players each have three pieces, which are set up at either end of a 3 by 3 board. Playing alternately, you can move a piece one or two squares in any direction (including diagonally) but must not pass over another piece. The winner is the first to get three pieces in a straight line (diagonals included) other than along the original starting line. Tsyanshidzi This is an ancient Chinese game. Alternately, players remove counters with the option of removing: (a) any number of counters from one pile. or (b) the same number of counters from each pile. The winner is the player who removes the last counter(s). Can you form a winning strategy for the first player? Alquerque (Africa) Two players each have 12 pieces, starting in the positions shown. Pieces can move along a line to an empty point. Pieces can be captured by being jumped over onto an empty point. More than one capture can be made in one move, and the direction of movement can also be changed. If a player misses a chance to capture an opponent’s piece, then the offending piece can be removed from the board. The winner is the first person to capture all of the opponent’s pieces. Dara (Nigeria) The board consists of 5 rows of 6 holes. Each player has 12 pieces, which are placed, in turn, into the holes. Once all of the pieces have been placed moves are made. A piece can be moved into an adjacent empty hole (not diagonally). When a line of 3 is formed the player removes one of the opponent’s pieces from the board. The game ends when a player is unable to make a line of 3 pieces. This game is played by the Dakarkari people using stones, pieces of pottery or shaped sticks. Exchange Kono (Korea) Each player has 8 pieces, with the starting position as shown. The players take it in turns to move a counter one space diagonally onto a black spot. The aim is to be the first to occupy the opponent’s starting positions. There are no jumps or captures. Fox and Geese (Iceland) This game was played by the Vikings. There are 13 geese and 1 fox. The geese start in the positions shown; the fox starts on any empty spot. Geese move first, along a line. The fox kills a goose by jumping over it to a vacant point. The geese win if they surround the fox. The fox wins if there are so many geese killed that it cannot be surrounded. Go Bang (Japan) In Japan the most popular game is Go, with professional players earning a lot of money. Go Bang is a simpler version of Go, arriving in England in 1885. Counters are placed alternately on the intersections of a 10x10 square board. The aim is to form 5 counters in a row in any direction. Kungser (Tibet) A battle game between 2 Princes and 24 Lamas. The Princes and Lamas are placed as shown. The first player (the Prince) can move a Prince one space or capture a Lama by jumping over it to an empty space beyond. The second player (the Lama) plays by placing a Lama on the board until all 24 have been used. Then the second player continues by moving Lamas on the board. The Prince wins if only 8 Lamas remain. The Lama wins if the Princes are trapped. Multiple captures are allowed. Mu Torere (New Zealand) Played by the Ngati Porou people, this is the only native Maori board game known. The board would have been marked on the ground with twigs or stones used as counters. Two players have 4 pieces (perepere) each placed on adjacent points of the star (kewai). The aim is to block the opponent from moving. The centre space is called the putahi. Moves can be made (a) from one kewai to an adjacent empty kewai, (b) from the putahi to a kewai, (c) from a kewai to the putahi as long as either one or both of the adjacent points is occupied by an enemy piece. Only one piece can occupy the same place at the same time. Jumping is not allowed. Shap Luk Kon Tseung Kwan One player is the general and the other controls the 16 soldiers. They can all move one step along any line in any direction. The general can enter the triangle at the top but the soldiers cannot. The general and the soldiers can capture. The general can capture two soldiers by moving to an empty point between them. Both soldiers are removed from the board. If the soldiers can position themselves so that they are directly beside the general on the same line the general is captured and loses. If the general is trapped inside the triangle he is captured and loses. (China) The Fifteen Game Draw a strip of cells and write in the numbers 1 to 9, as shown. 1 2 3 4 5 6 7 8 9 Players take turns claiming a number – perhaps by putting their initial(s) in that cell – but they must allow the number to be seen clearly. The winner is the first player to claim three numbers which add up to 15. The player may actually possess more than three numbers, but only three of them can be counted. This may look pretty uninteresting, but do not be deceived! (Make 15 analysis) Difficult to analyse? How many ways are there of making 15 with three numbers chosen from the above selection? This should reveal that some numbers are ‘better’ than others. Try setting out the nine numbers as a magic square and have players select their numbers from that by crossing them out with their own distinctive signs (like maybe an O and an X). What game are you really playing? Is it now easier to analyse? Does it even need analysing? This is a very practical example of an isomorphism. 4 9 2 3 5 7 8 1 6 Blox Draw a grid of any convenient size and shape. Players take turns putting their own distinctive mark (say an O and an X) in any cell. The only restriction is that no two cells which are side by side, touching along a common edge, may have the SAME type of mark in them. The winner is the last player who is able to make an allowed mark. End to End Draw a strip of any convenient number of cells. Place a counter in one end cell. O Players take turns advancing the counter towards the other end. In one turn a player may advance the counter 1, 2 or 3 cells. The winner is the player who actually moves the counter into the end cell. As a variation, it could be that the player who is forced to move into the end cell is the loser. Odd wins Draw a strip of 13 cells. O O O O O X X X X Players start at opposite ends. In turns players put their own marks in 1, 2 or 3 cells. Players must fill cells as they work from their own ends; no blanks may be left. When all the cells have been occupied, then the winner is the player who has made an ODD number of marks. Variation 1 Allow the marks to be placed anywhere, with the single restriction that, if 2 or 3 cells are filled in during one turn then they must be adjacent cells. (This is much more difficult to analyse.) Variation 2 Play it on a rectangular grid (provided it has an ODD number of cells). Again, multiple entries are only allowed in a set of cells connected by edges. In addition, an entry of 3 marks should be allowed only in a straight line of cells. Analysis now becomes possible only by computer. Capture the Numbers Write out the numbers 1 to 12. These are the numbers which are to be captured. Use 2 dice. Players take turns throwing the two dice and adding the two top numbers together to make a target value. A player may then capture either (a) one number which is equal in value to the target value; or (b) two numbers which add up to the target value. A number is captured by drawing a ring around it and identifying which player captured it. Sometimes no captures are possible. The game stops when it is clear that no more captures can be made. The winner is the player who has captured the most numbers. Diox Played like Noughts and Crosses on a 3 by 3 grid but, before each player has a turn he or she rolls a die to determine the mark which is to be made. If the die shows an odd number then the player has to put an O in the grid. If the die shows an even number then the player has to put an X in the grid. The winner is the first player to make a line of either Os or Xs. Wildox Played on a 4 by 4 grid. Players take turns putting either a O or an X in any cell. (Note that neither player has their own particular mark.) The winner is the first player to make a line of either 3 Os or 3 Xs in any direction (vertically, horizontally or diagonally). Winners or Losers Draw a strip of any convenient number of cells. Decide which player is to go first. The other player decides whether they shall play for winners or losers. They both start from the SAME end, filling in 1, 2 or 3 cells during their turn, but the filling in must be continuous; NO blanks may be left. Eventually all the cells must be filled in. The player who fills in the LAST cell(s) wins if they are playing winners loses if they are playing losers. The Game of Euclid This is based on the Euclidean algorithm for finding the highest common factor of two positive integers a, b. Each move changes the current pair of numbers a, b by subtracting some multiple of the smaller (say b) from the larger (say a) to get a new pair of numbers a-kb, b. Negative numbers are forbidden. The first player to produce a pair involving zero is deemed to be the winner. The art lies in choosing the right multiple of the smaller number to subtract each time. 27, 8 A 11, 8 B 3, 8 A 3, 5 B 3, 2 A 1, 2 B 1, 0 B wins this game. Could A have done better? Nine cards Two players have a pack of 9 cards. They take turns to remove 1, 2 or 3 cards from the pack; but a player must never remove the same number of cards as the previous player. The winner is the one who either takes the last card or leaves the other player with no valid move. What about different sized packs? Nine Men’s Morris Two players (black and white) each have 9 counters to place at any of the 24 points on the board. In part one, players play alternately placing men on the board. In part two, a turn is taken by moving a piece to an adjacent vacant point. The object is to form a “mill” or row of three – each time this is done, a player may remove one of the opponent’s men. The winner either blocks all his opponents men or reduces their number to two. Three Men’s Morris This simpler version of Nine Men’s Morris is played on a board with 9 points, with players having 4 counters each. Here the aim is simply to get three men in a line. In the cathedrals of Norwich, Canterbury, Gloucester, Salisbury and Westminster Abbey there are boards cut into the cloister seats doubtless to relieve the tedium of long services! Ko-no (China & India) One player has counters at A, B; the other at X,Y. A move must be to the vacant space – a player loses when he is blocked. Teachers may be surprised at the problem solving strategies which this simple game develops in young children. A B X Y Mathematical Recreations For example: A wolf, a goat and a cabbage must be moved across a river in a boat holding only one besides the ferryman. How must he carry them across so that the goat shall not eat the cabbage, nor the wolf the goat? From: Problems for the Quickening of the Mind by Alcuin of York (c. 775).