Game 100 In 1 Jar
Two players, Amy and Beth, play the following game with a jar containing 100 pennies. The players take turns; Amy goes first. Each time it is a player's turn, she takes between 1 and 10 pennies out of the jar. The player whose move empties the jar loses.
game 100 in 1 jar
Section: Strategy in game theory means the moves and actions played by the players associated with the game. The strategies chosen by the players are done to maximize their benefits from the game. The players prepare a plan of action before formulating strategies.
The above game establishes the rules for Bath and Amy in such a way that the player who empties the jar loses the game. In this case, Bath and Amy will devise their strategy so that they leave the opponent player with one penny in the jar. Amy plays first and removes one penny leaving 99 pennies behind. Now it is not important how many are removes by Beth. As per the equation Amy will remove 11- whatever is removed by Beth. In the end, there are 11 pennies left in the jar Amy. No matter how many pennies are removes by Beth, Amy will be the last one to remove penny/ies from the jar and will lose the game. Hence the game has a second-mover advantage.
Nim is a mathematical game of strategy in which two players take turns removing (or "nimming") objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object.
Nim is typically played as a misère game, in which the player to take the last object loses. Nim can also be played as a normal play game whereby the player taking the last object wins. This is called normal play because the last move is a winning move in most games, even though it is not the normal way that Nim is played. In either normal play or a misère game, when the number of heaps with at least two objects is exactly equal to one, the player who takes next can easily win. If this removes either all or all but one objects from the heap that has two or more, then no heaps will have more than one object, so the players are forced to alternate removing exactly one object until the game ends. If the player leaves an even number of non-zero heaps (as the player would do in normal play), the player takes last; if the player leaves an odd number of heaps (as the player would do in misère play), then the other player takes last.
At the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim. From May 11, 1940, to October 27, 1940, only a few people were able to beat the machine in that six-week period; if they did, they were presented with a coin that said Nim Champ. It was also one of the first-ever electronic computerized games. Ferranti built a Nim playing computer which was displayed at the Festival of Britain in 1951. In 1952 Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. L. Maxon Corporation, developed a machine weighing 23 kilograms (50 lb) which played Nim against a human opponent and regularly won. A Nim Playing Machine has been described made from TinkerToy.
The normal game is between two players and is played with three heaps of any number of objects. The two players alternate taking any number of objects from any one of the heaps. The goal is to be the last to take an object. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object.
The practical strategy to win at the game of Nim is for a player to get the other into one of the following positions, and every successive turn afterwards they should be able to make one of the smaller positions. Only the last move changes between misere and normal play.
When played as a misère game, Nim strategy is different only when the normal play move would leave only heaps of size one. In that case, the correct move is to leave an odd number of heaps of size one (in normal play, the correct move would be to leave an even number of such heaps).
These strategies for normal play and a misère game are the same until the number of heaps with at least two objects is exactly equal to one. At that point, the next player removes either all objects (or all but one) from the heap that has two or more, so no heaps will have more than one object (in other words, so all remaining heaps have exactly one object each), so the players are forced to alternate removing exactly one object until the game ends. In normal play, the player leaves an even number of non-zero heaps, so the same player takes last; in misère play, the player leaves an odd number of non-zero heaps, so the other player takes last.
Theorem. In a normal Nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is not zero. Otherwise, the second player has a winning strategy.
In another game which is commonly known as Nim (but is better called the subtraction game), an upper bound is imposed on the number of objects that can be removed in a turn. Instead of removing arbitrarily many objects, a player can only remove 1 or 2 or ... or k at a time. This game is commonly played in practice with only one heap.
Bouton's analysis carries over easily to the general multiple-heap version of this game. The only difference is that as a first step, before computing the Nim-sums we must reduce the sizes of the heaps modulo k + 1. If this makes all the heaps of size zero (in misère play), the winning move is to take k objects from one of the heaps. In particular, in ideal play from a single heap of n objects, the second player can win if and only if
A similar version is the "100 game": Two players start from 0 and alternately add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g., 01, 12, 23, 34,...) and control the game by jumping through all the numbers of this sequence. Once a player reaches 89, the opponent can only choose numbers from 90 to 99, and the next answer can in any case be 100.
In Grundy's game, another variation of Nim, a number of objects are placed in an initial heap and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of 5+1 or 4+2, but not 3+3. Grundy's game can be played as either misère or normal play.
Greedy Nim is a variation wherein the players are restricted to choosing stones from only the largest pile. It is a finite impartial game. Greedy Nim Misère has the same rules as Greedy Nim, but here the last player able to make a move loses.
Let the largest number of stones in a pile be m and the second largest number of stones in a pile be n. Let pm be the number of piles having m stones and pn be the number of piles having n stones. Then there is a theorem that game positions with pm even are P positions. This theorem can be shown by considering the positions where pm is odd. If pm is larger than 1, all stones may be removed from this pile to reduce pm by 1 and the new pm will be even. If pm = 1 (i.e. the largest heap is unique), there are two cases:
A generalization of multi-heap Nim was called "Nim k \displaystyle _k " or "index-k" Nim by E. H. Moore, who analyzed it in 1910. In index-k Nim, instead of removing objects from only one heap, players can remove objects from at least one but up to k different heaps. The number of elements that may be removed from each heap may be either arbitrary or limited to at most r elements, like in the "subtraction game" above.
Building Nim is a variant of Nim wherein the two players first construct the game of Nim. Given n stones and s empty piles, the players, alternating turns, place exactly one stone into a pile of their choice. Once all the stones are placed, a game of Nim begins, starting with the next player that would move. This game is denoted BN(n,s).
Candy Nim is a version of normal play Nim in which players try to achieve two goals at the same time: taking the last object (in this case, candy) and taking the maximum number of candies by the end of the game.
Whenever this happens, I pull out the 100 Game. This game can be taught in two minutes but always keeps my students occupied for at least twenty as they battle each other and develop strategies to try to unseat me, the 100 Game World Champion.
When I introduce this game to my students, I like to be extremely arrogant and play up the World Champion angle. I ask for a volunteer to play against me as the official First Loser of the 100 Game. I require that each student win three games against their classmates before they earn the opportunity to lose to me. I even trash talk during the game, which is not usually a part of my typical math class.All of this is fuel for those couple of students who desperately want to beat me and are willing to go home, sit down with a piece of paper, to finally figure out how to beat me. Every year I have eventually been unseated as World Champion, which is one of the most fun days of the year.
The 100 Game is clearly mathematical, but on two levels. Most simply, it's a great game for young kids to practice subtraction. A 2nd grader would feel comfortable playing this game, which gives some structure and reason for the repeated subtraction practice. When I play a kid, I make them keep track of both players' moves, which doubles the practice. Each game is the practice equivalent of a classic subtraction worksheet, but your kids won't complain nearly as much.