analysis and logic as well as some basic knowledge about numbers, are key to the study of the different strategies of popular games such as chess , playing cards, craps and many more, this time talk of a special family of games of strategy, known as NIM. These can be used as a good methodological tool in the classroom or simply to bring our children to mathematics in our own homes.
The origin of this type of games are located several millennia ago, and practiced in the steppes of central Asia. It is not really a board game, because originally it was played on the ground, with objects found in the same place as rocks, sticks, grains, among others. On these games, you can see [2] and [3] , especially interesting article on teaching math games of Miguel de Guzman, which displays a precise and rigorous historical The various types of games and its utility in education, beyond the mere recreational activity.
In different games we'll see, when we speak of objects relates to toothpicks, beans, coins, among others. A good strategy to solve the games NIM is to study the simpler games to try to discover the pattern that determines which of the two players wins (The beginning or the second).
They lend themselves to discuss properties on the numbers, they can enjoy discovering the winning strategy and are fit to play for all audiences. The rules for these games can be modified by changing the number of objects at the beginning, the number of objects that can be taken on every play or if you take the last object wins or loses. Starting, of course, on the premise that they are playing their best, always look for the best hand possible.
Here are some variations NIM game, taken from [2]. All games offered are for two people necessarily have a winner and all have a finite number of moves. For these games, the game turn alternates.
Game 1. Start with ten objects. In his turn may take one or two items. The player who removes the last object wins.
Game 2. Start with fifteen objects. On your turn you can take one, two or three objects. In analyzing the game, the player who holds a pair of objects.
Game 3. Start with three sets of objects. One of three, a five and a third group of seven objects. On your turn you can take a whole group or only a part of it. The player who takes the last purpose of the latter group wins.
Game 4. Place eight objects in a row. When is your turn to take one or two objects, but if you take two, they must be adjacent. Whoever takes the last object wins.
NIM
A typical player can not move, for example, when there are more pieces to take, is a defeat, however, this rule can be changed and thereby obtain significant changes in planning strategy.
Game 5. Same as above, but the person can not move wins, ie you are playing an inverted version of the previous game, except the player taking the last lost object.
Game 6. Start with a group of thirteen objects. In turn, separate the group into two unequal heaps, ie with different numbers of objects. If you can not, you lose.
For example, in my turn I can separate the group of 13 into two groups of six and seven. Now it's your turn. You can separate the group of seven groups of six and one, or five-two or three and four. You can also separate group of six in one and five or two and four. It is not possible to separate it into three and three. The game is over when there are only groups of two and an object.
The example below is made thinking of currencies, however, this game can use any object that has two different faces, face call them A and B.
Game 7. Start with a row of ten coins with the A side up. In turn, you must turn a coin, ie, face B up, and also has the option of turning any other currency that is to the left of the first to invest. If you can not play, that is, all currencies are in the B-side up, loses.
Both players should be looking at the row of coins from the same side, so that the direction "left" is the same. Let's review the rules of this game. In your turn you can flip a coin or two. The second currency, the optional-can be turned to face to get up.
For example, if I flip the coin five, you can turn seven and flip the coin flip the coin also five to return to stay with the A side up. Note that the last coin on the right, the ten, can only be turned once, since it has no currency on the right.
Game 8. Consider the following game: With an initial amount of 40 stones, players can, at every turn, remove 1, 2, 3, 4 or 5 stones at will and who wins takes the last stone. If we assume that A and B play and you start to play A. What is the winning strategy for A?
In a forthcoming contribution, I will continue with several sets of type NIM, related to some interesting properties of integers.
The origin of this type of games are located several millennia ago, and practiced in the steppes of central Asia. It is not really a board game, because originally it was played on the ground, with objects found in the same place as rocks, sticks, grains, among others. On these games, you can see [2] and [3] , especially interesting article on teaching math games of Miguel de Guzman, which displays a precise and rigorous historical The various types of games and its utility in education, beyond the mere recreational activity.
In different games we'll see, when we speak of objects relates to toothpicks, beans, coins, among others. A good strategy to solve the games NIM is to study the simpler games to try to discover the pattern that determines which of the two players wins (The beginning or the second).
They lend themselves to discuss properties on the numbers, they can enjoy discovering the winning strategy and are fit to play for all audiences. The rules for these games can be modified by changing the number of objects at the beginning, the number of objects that can be taken on every play or if you take the last object wins or loses. Starting, of course, on the premise that they are playing their best, always look for the best hand possible.
Here are some variations NIM game, taken from [2]. All games offered are for two people necessarily have a winner and all have a finite number of moves. For these games, the game turn alternates.
Game 1. Start with ten objects. In his turn may take one or two items. The player who removes the last object wins.
Game 2. Start with fifteen objects. On your turn you can take one, two or three objects. In analyzing the game, the player who holds a pair of objects.
Game 3. Start with three sets of objects. One of three, a five and a third group of seven objects. On your turn you can take a whole group or only a part of it. The player who takes the last purpose of the latter group wins.
Game 4. Place eight objects in a row. When is your turn to take one or two objects, but if you take two, they must be adjacent. Whoever takes the last object wins.
NIM
A typical player can not move, for example, when there are more pieces to take, is a defeat, however, this rule can be changed and thereby obtain significant changes in planning strategy.
Game 5. Same as above, but the person can not move wins, ie you are playing an inverted version of the previous game, except the player taking the last lost object.
Game 6. Start with a group of thirteen objects. In turn, separate the group into two unequal heaps, ie with different numbers of objects. If you can not, you lose.
For example, in my turn I can separate the group of 13 into two groups of six and seven. Now it's your turn. You can separate the group of seven groups of six and one, or five-two or three and four. You can also separate group of six in one and five or two and four. It is not possible to separate it into three and three. The game is over when there are only groups of two and an object.
The example below is made thinking of currencies, however, this game can use any object that has two different faces, face call them A and B.
Game 7. Start with a row of ten coins with the A side up. In turn, you must turn a coin, ie, face B up, and also has the option of turning any other currency that is to the left of the first to invest. If you can not play, that is, all currencies are in the B-side up, loses.
Both players should be looking at the row of coins from the same side, so that the direction "left" is the same. Let's review the rules of this game. In your turn you can flip a coin or two. The second currency, the optional-can be turned to face to get up.
For example, if I flip the coin five, you can turn seven and flip the coin flip the coin also five to return to stay with the A side up. Note that the last coin on the right, the ten, can only be turned once, since it has no currency on the right.
Game 8. Consider the following game: With an initial amount of 40 stones, players can, at every turn, remove 1, 2, 3, 4 or 5 stones at will and who wins takes the last stone. If we assume that A and B play and you start to play A. What is the winning strategy for A?
In a forthcoming contribution, I will continue with several sets of type NIM, related to some interesting properties of integers.
"God does not play dice"
Albert Einstein
References [1] Guzman, Miguel. games in teaching math , Spain.
[2] Erickson, Timothy. Bored of the typical NIM?, Proceedings of the First Festival of Mathematics, Costa Rica, 1998.
[3] T. Murillo Manuel and Gonzalez A. Fabio. Number Theory , Editorial Tecnológica, Costa Rica, 2006.
[4] T. Murillo Manuel et al. A mosaic chess, education, mathematics and the Internet, Costa Rica, 2001.
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