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In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n , one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0 .
The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game. [7] [8] Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. [9]
This article gives an example of a zero-sum game that has no value. It is due to Sion and Wolfe. [1] Zero-sum games with a finite number of pure strategies are known to have a minimax value (originally proved by John von Neumann) but this is not necessarily the case if the game has an infinite set of strategies. There follows a simple example ...
Integer overflow can be demonstrated through an odometer overflowing, a mechanical version of the phenomenon. All digits are set to the maximum 9 and the next increment of the white digit causes a cascade of carry-over additions setting all digits to 0, but there is no higher digit (1,000,000s digit) to change to a 1, so the counter resets to zero.
returns the nearest integer, rounding away from zero in halfway cases nearbyint: returns the nearest integer using current rounding mode rint lrint llrint: returns the nearest integer using current rounding mode with exception if the result differs Floating-point manipulation functions frexp: decomposes a number into significand and a power of ...
This name derives from the shape of the marks in the game (i.e the X and O); "nought" is another name for the number zero, while "cross" refers to the X shape. Sometimes, tic-tac-toe (where players keep adding "pieces") and three men's morris (where pieces start to move after a certain number have been placed) are confused with each other.
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In computational complexity theory, the 3SUM problem asks if a given set of real numbers contains three elements that sum to zero. A generalized version, k-SUM, asks the same question on k elements, rather than simply 3. 3SUM can be easily solved in () time, and matching (⌈ / ⌉) lower bounds are known in some specialized models of computation (Erickson 1999).