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Bookworm is a general name for any insect that is said to bore through books. [ 1 ] [ 2 ] The damage to books that is commonly attributed to "bookworms" is often caused by the larvae of various types of insects, including beetles , moths , and cockroaches , which may bore or chew through books seeking food.
The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4). [3] Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular. [4]
An unbiased random walk, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely. A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair.
This is the basis for constructing squares that display some information (e.g. birthdays, years, etc.) in the square and for creating "reversible squares". For example, we can display the number π ≈ 3.141 592 at the bottom row of a 4×4 magic square using the Graeco-Latin square given above by assigning (α, β, γ, δ) = (10, 0, 90, 15) and ...
Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is an example of a dihedron. The Schläfli symbol is {4,2}. Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles ...
In Figure 6, for example, which is magic on rows and columns only, the 16 pieces form a so-called self-tiling tile set. Such a set is defined as any set of n distinct shapes, each of which can be tiled by smaller replicas of the complete set of n shapes. [14] A second example is Figure 4, which is a so-called 'self-interlocking' geomagic square.
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Proof without words that all centered octagonal numbers are odd squares. Squares of even numbers are even, and are divisible by 4, since (2n) 2 = 4n 2. Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when ...