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Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a 2 b 3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.
The Square-1 (full name "Back to Square One") or alternatively, "Cube 21", was invented by Karel Hršel and Vojtěch Kopský in 1990.Application for a Czechoslovak patent was filed on 8 November 1990, then filed as a "priority document" on January 1, 1991.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
[1] Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem ). Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem ; see Fermat's right triangle ...
The position of this cell is the extreme foreground of the 4th dimension beyond the position of the viewer's screen. 4-cube 3 4 virtual puzzle, rotated in the 4th dimension to show the colour of the hidden cell. 4-cube 3 4 virtual puzzle, rotated in normal 3D space. 4-cube 3 4 virtual puzzle, scrambled. 4-cube 2 4 virtual puzzle, one cubie is ...
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Skewb, a cube-shaped puzzle added in 2014 as an official WCA event due to its growing popularity and the ease of its regulations. [28] Square-1, a cubed puzzle that changes shape as it is solved; Rubik's Clock, a double-sided circle-shaped puzzle with 9 clock faces on it that is considered solved when all clock hands are in the 12 o'clock position.
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