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In game theory, Silverman's game is a two-person zero-sum game played on the unit square. It is named for mathematician David Silverman. It is played by two players on a given set S of positive real numbers. Before play starts, a threshold T and penalty ν are chosen with 1 < T < ∞ and 0 < ν < ∞.
The original version of 24 is played with an ordinary deck of playing cards with all the face cards removed. The aces are taken to have the value 1 and the basic game proceeds by having 4 cards dealt and the first player that can achieve the number 24 exactly using only allowed operations (addition, subtraction, multiplication, division, and parentheses) wins the hand.
A two-player version of Taxman, known simply as The Factor Game, was described in an article for the November 1973 issue of The Arithmetic Teacher, a publication of the National Council of Teachers of Mathematics. [26] The article was later reprinted in the 1975 anthology Games and Puzzles for Elementary and Middle School Mathematics. [4]
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We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c.. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0)Each equation follows by definition [A1]; the first with a + b, the second with b.
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