Integer Length Research Articles

Orthogonal Bases of ℝ3 with Integer Coordinates and Integer Lengths

(1989). Orthogonal Bases of ℝ3 with Integer Coordinates and Integer Lengths. The American Mathematical Monthly: Vol. 96, No. 1, pp. 49-53.

New binary Euclidean algorithms

Two new binary Euclidean algorithms to calculate the greatest common divisor are given. An exhaustive search for all odd integers of moderate length shows that these algorithms use fewer iterations on the average than that the two presently known algorithms.

Avoiding-sequences with minimum sum

A circular sequence of positive integers of length n is a sequence of n terms in which the first and last are considered consecutive. For x≤ n a positive integer, such a sequence is x-avoiding if no set of consecutive terms sums to x. We show that an x-avoiding circular sequence of length n satisfies a 1+ a 2+⋯+ a n ≥2 n, and give a simple necessary and sufficient condition for equality. Minimizing sequences are exhibited when the minimum sum is known.

Percolational and fractal property of random sequential packing patterns in square cellular structures.

The problem for random packing or filling has been treated in many fields. In this paper we form random sequential packing patterns, filling metallic squares with integer length a on insulator substrates divided into square unit cells. We investigate the percolational and fractal property of the packing patterns, and clarify that the maximum critical percolation length ${a}_{c}$ (=3 units length) of the packed squares is presented for the insulator-to-metal transition to take place on the random sequential packing textures. When a>${a}_{c}$, no insulator-to-metal transition occurs even at the saturation coverage where no more squares can be filled without any overlap. However even in such patterns with a>${a}_{c}$, the large percolation clusters possess fractal properties, and the fractal dimensions equal 1.94.

Randon sequential packing fraction in square cellular structures

Hitherto the problem of random sequential packing has been treated in continuum space. In a previous paper we introduced random sequential packing into the square cellular structure, where squares with integer length are inserted at random without any overlap into the cells of a square substrate divided into square unit cells. To the packing we applied two methods A and B, which were not distinguished in the packing of continuum space. In A, contact between the packed squares is permitted, and in B such contact is forbidden. In the previous paper we independently obtained the packing fractions of A and B, and did not notice the relation between them. Here, however, we make clear the relation between the packing fractions of A and B and continuum space.

Random sequential packing in square cellular structures

The problem of random sequential packing has been studied in statistical physics, chemical physics and biophysics. The problem has been treated mainly in the continuum or lattice space, while little attention has been paid to the packing problem in a cellular structure. As a first step to random filling in cellular structures, here the author treats the problem in the square cellular structure, where squares with integer length a are inserted at random without any overlap into the cells of a square divided into square unit cells. In such packing problem, two methods A and B, which are not distinguished in the continuum space, are applied to filling squares. In A any contact among the packed squares is permitted, and forbidden in B. The author calculates the packing fractions of A and B against a by computer simulation, and clearly shows that the packing fraction of A is much larger than that of B when a is small. As a becomes large, the two packing fractions approach that of the continuum space, respectively, from the upper and lower side.

A Bug's Shortest Path on a Cube

This can be readily obtained by identifying each shortest path either as a selection of m objects from m + n objects or as a permutation of m + n objects consisting of exactly m of the symbols E (east) and n of the symbols N (north). Now one naturally wonders about the 3-dimensional analogue of this problem. Imagine for example, that a bug situated at one corner of an n X n X n Rubik's cube wants to reach the opposite corner of the opposite face in such a way that its path consists of line segments of integer length each of which is parallel to one of the axes. What is the number of such shortest paths? If the bug can chew its way through the interior of the cube, then the same argument used in the 2-dimensional version immediately gives the answer (3 n) !/( n !)3. But if the bug can only crawl on the surface of the cube and because the surface is slippery, must crawl along the edges or the grooves between the small cubes, to determine the number, f (n), of such shortest paths turns out to be a much harder problem than the 2-dimensional version. Though it is obvious that f(1) = 6, to compute the value of f(2) by brute force might be quite a challenging job. (Any one with 2 x 2 x 2 Rubik's cube is encouraged to give it a try.) To determine the value of f(3), with or without the use of an ordinary Rubik's cube, seems to be a formidable task. For the interest of the readers, we will not reveal these two values until the end of the article. To answer this question, we consider the more general problem in which the cube is replaced by a rectangular box. Let Q be the I X m X n rectangular prism in space with vertices at A = (0, 0, 0), B = (1, 0, 0), C=(l,m,O), D=(O,m,0), E=(0,m,n), F=(0,0,n), G=(l,0,n) and H=(l,m,n). See FIGURE 1, where 1, m, and n are positive integers. A bug wants to go from A to H on the surface of Q in such a way that its path consists of line segments of integer lengths each of which is parallel to one of the axes. What is the number f (1, m, n) of different shortest paths? (It is clear that the length of such a path is I + m + n.) This is question (1) asked in [1].

Integer-sided triangles with an angle of 60°

We are looking for all triangles with sides of integer lengths a, b, and c say, with an angle of 60° opposite the side of length a.

SOL: an exercise in defining a language

A language for operation on a stack (SOL) is defined formally, also from the semantic point of view. The semantics is an input-output semantics which uses functions from sequences (of integers) of indefinite length to sequences (of integers) of indefinite length as models of the computation on a stack. The programs in SOL are interpreted in recursive systems of functional definitions to which fixed point theory easily applies in order to evaluate the functions computed by the programs. The language has been implemented and used for didactic purposes.