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Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone, [2] and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. [3]
A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. [4] For example, the Latin square above is not reduced because its first column is A, C, B rather than A, B, C. Any Latin square can be reduced by permuting (that is, reordering) the rows and columns ...
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as ...
Several packing problems concern squares. For instance, the problem of square packing in a square or circle asks how many unit squares can fit without overlap into a larger square or circle of a given size. Alternatively and equivalently, one may ask to minimize the size of a larger square or circle that contains a given number of unit squares ...
Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares) The British flag theorem for rectangles equates two sums of two squares; The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. [1]
The congruum problem may be solved by choosing two distinct positive integers and (with >); then the number () is a congruum. The middle square of the associated arithmetic progression of squares is (+), and the other two squares may be found by adding or subtracting the congruum.
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