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42 is a pronic number, [1] an abundant number [2] as well as a highly abundant number, [3] a practical number, [4] an admirable number, [5] and a Catalan number. [6]The 42-sided tetracontadigon is the largest such regular polygon that can only tile a vertex alongside other regular polygons, without tiling the plane.
A scrambled Rubik's Cube. An algorithm to determine the minimum number of moves to solve Rubik's Cube was published in 1997 by Richard Korf. [10] While it had been known since 1995 that 20 was a lower bound on the number of moves for the solution in the worst case, Tom Rokicki proved in 2010 that no configuration requires more than 20 moves. [11]
The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.
Construction of Q 3 by connecting pairs of corresponding vertices in two copies of Q 2. The hypercube graph Q n may be constructed from the family of subsets of a set with n elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element.
Completing the cube is a similar technique that allows to transform a cubic polynomial into a cubic polynomial without term of degree two. More precisely, if a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d}
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Semi-log plot of solutions of + + = for integer , , and , and .Green bands denote values of proven not to have a solution.. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.