[5] **viXra:1307.0095 [pdf]**
*replaced on 2014-01-11 12:49:08*

**Authors:** Marco Ripà

**Comments:** 13 Pages.

A generalization of Ripà’s square spiral solution for the nXnX…X n points upper bound problem. Additionally, we provide a non-trivial lower bound for the k-dimensional n_1Xn_2X…Xn_k points problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we provide a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

**Category:** Combinatorics and Graph Theory

[4] **viXra:1307.0055 [pdf]**
*submitted on 2013-07-11 11:56:55*

**Authors:** John Frederick Sweeney

**Comments:** 20 Pages.

In Vedic Physics, the combinatorial process by which counts are counted closely resembles the binomial triangle, or Pascal's Triangle, which was known to Ancient India and China as Mount Meru or Zhang's triangle. This is not accidental, but instead reflects the numerical structure of the universe, as the pattern repeats itself in many types of numbers, including Clifford Algebras, Octonions, the Exceptional Lie Algebras, the Magic Triangle, and Barnes - Wall Lattices. Previous papers have discussed details of the counts; this paper focuses on the triangular relationships between numbers, especially those noted by Tony "Frank" Smith.

**Category:** Combinatorics and Graph Theory

[3] **viXra:1307.0052 [pdf]**
*submitted on 2013-07-10 04:56:05*

**Authors:** John Frederick Sweeney

**Comments:** 11 Pages.

Matter may exist in one of three states in the universe. The interactions between these states of matter add up to fifty types of change. Causes of change include: the spectrum of inter-changed states, due to imbalance, failure to synchronize, balance and coherent synchronization, caused by the interplay of three modes of matter, which sum up to 50 (order of powers). This paper describes the three states of matter and how their interactions combine to create fifty types of change.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1307.0041 [pdf]**
*submitted on 2013-07-08 10:59:13*

**Authors:** John Frederick Sweeney

**Comments:** 9 Pages.

Decoding of Vedic literature reveals three states of matter, and the basic count of matter as modulated by the Golden Section. Thus, all forms of nature follow the contours of the Golden Section in their growth and development. Provided here is a basic explanation of how the counts occur within the three separate states of matter, and a new formation of Time. These concepts form the basis of the Qi Men Dun Jia Model, with its incorporation of the icosahedron and its 60 stellated permutations.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1307.0021 [pdf]**
*replaced on 2013-09-05 19:08:16*

**Authors:** Marco Ripà, Pablo Remirez

**Comments:** 11 Pages. This is the second version of the paper in Italian that I already submitted a few months ago. It has been entirely written in English.

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. One of the earliest appearances is in “Cyclopedia of Puzzles” by Sam Loyd in 1914. Here is a review of the generic solution of the problem of the 9 points spread to n^2 points. Basing it on a specific pattern, we show that any nxn (for n ≥ 5) points puzzle can also be solved ‘Inside the Box’, using only 2∙n − 2 straight lines (connected at their end-points), through the square spiral method. The same pattern is also useful to “bound above” the minimal number of straight lines we need to connect n^k points in a k-dimensional space, while to “bound below” the solution of the nxnx…xn puzzle we start from a very basic consideration.

**Category:** Combinatorics and Graph Theory