REGENERATIVE MATHEMATICS

### Regenerative Mathematics

November, 2013

Distributive Regeneration of Ordered System is a phenomenon which occurs when a given system of order comprising a number of elements that are grouped into two or more groups is subjected to a Logico-Sequential Distribution for the purpose of regeneration. Regenerative mathematics otherwise known as Kola Analysis is the mathematical study of distributive regeneration of ordered system. The concept of regenerative mathematics came into emergence when it was observed that a set of objects that were grouped into two or more columns and rows when subjected to Logico-sequential distribution returned to their original arrangement before being distributed after certain distribution number.

It was further confirmed that different arrangements of these objects have their own regenerative distribution numbers. It was this observation that led to the First Law of Logico-Sequential Distribution and it states that every ordered system comprising elements (objects) which can be grouped into two or more columns and rows whether a simple or multiple dimensions has a Regenerative Distribution Number (t), if the elements (objects) in the ordered system are compelled to undergo a series of transposition by Logico-Sequential Distribution impressed on them.

In other words, when different sets of ordered system comprising elements that are grouped into two or more rows and columns in a simple dimension are made to undergo Logico-Sequential Distribution, their Regenerative Distribution Numbers (t) differ, obeying specific mathematical proportionalities and formulas which could be linear, quadratic, inductive and sequential.

Regenerative Distribution Number (t) and Second to None Regenerative Distribution Number (s)

Regenerative Distribution Number (t) is defined as the number of transformed distributions it takes for a regeneration of the original state of an ordered system.

Second to None Distribution to the Regenerated d0 in the distribution cycle is denoted by (ds) where s = t - 1 (the Second to None Regenerative Distribution Number or the last transposed version distribution number of the starting arrangement).

The value of last transposed version distribution number (s) of the starting arrangement (d0) is very crucial to CMIG tricks because when you compare both the vertical and horizontal position ranking values of any card at starting arrangement (d0) with its vertical and horizontal position ranking values at ds (the second to none distribution to the regenerated d0), you would assert that the position ranking values of cards at d0 are interchanged vertically and horizontally respectively at ds.That is to say, if a card at d0 has its horizontal position ranking value (Phd0) = 11, at ds the same card would have its (Pvds) = 11. Therefore if a performer instructed a spectator to put his chosen card at the horizontal position ranking value 11th at the starting arrangement (d0) at ds (the second to none distribution to the regenerated d0) that particular chosen card by the spectator would be at the vertical position ranking value 11th. That is to say when all the cards are gathered into a single column, the spectator's chosen card would be at 11th position counting from top downward.

Transposition is the process of causing two or more things to change their places or positions. Using the Logico-sequential distribution, when elements (cards) are being distributed, they change their positions from one distribution to another until they finally reach the last transposed version distribution of the starting arrangement (d0) denoted by ds. In the distribution series, the preceded distribution is the transposed version of the subsequent distribution e.g considering this distribution.

Table 1

d0 d1
12345 16116117
678910 12721813
1112131415 8319149
1617181420 42015105
d2 = ds d3 = dt
48121620 12345
37111519 678910
26101418 1112131415
1591317 1617181920

From Table 1, d1 is the transposed version of d0, d2 = ds, is the last transposed version of d0
d3 = dt, is the regenerated version of d0, d2 = dt - d1= ds
Also there is a midpoint transposed version termed dsm which has mathematical relation with ds (the last transposed version of d0) e.g considering the below distribution.

Table 2

d0 d1 d2
12345 61728 36914
678910 394105 710258
d3 d4 = dsm d5
731062 97531 109876
95184 108642 54321
d6 d7 d8
510493 852107 48159
82716 41963 261037
d9 = ds d10 = dt
246810 12345
13579 678910

So the distribution goes thus,
d0, d1, d2, d3, d4 = dsm, d5, d6, d7, d8, d9 = ds, d10 = dt (regenerated d0)
d4 = dsm is the midpoint transposed version & the transposed version of d5
d9 = dt-d1 is the last transposed version of d0 or d10 (regenerated d0)
The relationship between dsm and ds
The mathematical relationship between dsm and ds is given as below.
ds =2dsm + d1........................(1)
Where s = t - 1.......................(2)
(i)For examples, if dsm = d1
Then ds = 2 x d1 + d1 = d3
(ii) if dsm = d2
Then ds = 2 x d2 + d1 =d5
(iii) if dsm = d32
Then ds = 2 x d32 + d1 = d65
The calculation of regenerative distribution number (t) and that of last transposed distribution number (s) is very pertinent to card mathematical intelligence games (CMIG). Some of the mathematical proportionalities and formulas that can be used to calculate regenerative distribution number would be treaed. You may decide to use any number of cards for CMIG tricks once you know how to calculate regenerative distribution number of your chosen arrangement. Another area that is also interesting is that through mathematical formulas you can monitor the positional change of any element (card) in the distribution cycle.
That is to say a formula for calculating the subsequent positional change of any element in the distribution cycle would be used in calculating both vertical and horizontal position ranking values of elements (cards) in the distribution cycle. This aspect of Kola Analysis is called Positiomatics. Some aspects of the Analysis of Distributive Regeneration of Ordered System in Simple dimension brought about card mathematical intelligence games.

### KOLA ANALYSIS

Kola Analysis is the mathematical analysis of a logical-mathematical intelligence phenomenon called Distributive Regeneration of Ordered System. This involves derivation of formulas and proportionalities governing the phenomenon by gathering and analyzing data, recognizing patterns, handling of logical thinking through mathematical manipulative and critical thinking skills and solving intelligence questions.
Kola Analysis is divided into four.

1. Analysis of Distributive Regeneration of Ordered System: This is divided into two.
(A) Analysis of Distributive regeneration of ordered System in simple dimension. This is further categorized as follows:
(i) Power Inductive Distribution,
(ii) Column Inductive Distribution,
(iii) Sequential graphical Distribution,
(v) Exceptional Distributions.
(B) The analysis of Distributive Regeneration of Ordered System in Multiple Dimension

2. Positiomatics: This is divided into two as follows.
(i) Simple Dimension Positiomatics
(ii) Multi-Dimension Postiomatics

3. Complexity Emergence in Distributive Regeneration of Ordered System: This is the analysis of effect of removal of elements in an ordered system on Regenerative Distribution Number (t) and Positiomatics when subjected to Logico-Sequential Distribution.

4. Cryptography and application of Distributive regeneration of Ordered System:This involves research on the real world application of Kola Analysis in computer programming, intelligence game development and software development that have special features in encryption, security and defence, computer simulation, military and civil intelligence and everyday work applications.

According to Taylor (2010), a set of ordered systems where the total number of elements n (E) could be expressed in form of arrangement Ce[Ce - 2], the overall formula connecting this arrangement and its even regenerative distribution number (te) is given as n (E) = Ce [Ce - 2] = 1/4 [te]2 - 1...........(1)
where Ce = Even number of columns in the ordered system, r = [Ce - 2] = Even number of rows and te = Even Regenerative Distribution Number.

There are series of formulas under the quadratic distribution aspect of distributive regeneration of ordered system but for the purpose of this book, the above formula and its derivative would be used for illustrations.Since t = s + 1
te = so + 1, so is the odd last transposed version number of the starting distribution or arrangement.

Ce[Ce - 2] = 1/4 [so + 1]2 - 1
Ce[Ce - 2] = 1/4 [so]2 + 1/2 so - 3/4......................(2)

Illustrations
(1) In a football match opening ceremony, 80 students are needed to make a parade to grace the occasion. They are grouped into 10 columns with each column having 8 students and if their parade mechanism obeys Logico-Sequential Distribution. Calculate the Regenerative Distribution Number (t) of the arrangement.
Solution
The total number of students, n (E) = 80, Number of columns = 10
Since 10 x 8 can be expressed in terms of Ce[Ce - 2]
Ce[Ce - 2] = 1/4 [te]2 - 1
10 x 8 = 1/4 [te]2 - 1
t = -18 or 18
Since we are dealing with concrete things te = 18
The Regenerative Distribution Number (t) of the arrangement = 18

(2) Calculate the last transposed version distribution number of the starting arrangement, if 24 cards are grouped into 6 columns with each column having 4 cards.
Solution
n (E) = 24, Since 6 x 4 can be expressed in terms of Ce[Ce - 2]

Ce[Ce - 2] = 1/4 [so]2 + 1/2 so - 3/4
6 x 4 = 1/4 [so]2 + 1/2 so - 3/4
24 x 4 = [so]2 + 2 so - 3
[so]2 + 2 so - 99 = 0
so = - 11 or 9
Since we are dealing with concrete things so = 9. The last transposed version distribution number of the starting arrangement = 9.

Problems
1. Calculate the regenerative distribution number (t), if 80 cards are grouped into 10 columns with each column having 8 cards.

2. Calculate the regenerative distribution number (t), if 99 cards are grouped into 9 columns with each column having 11 cards.

3. If a student grouped 399 cards into 19 columns with each column having 21 cards. Calculate (i) the last transposed version distribution number of the starting distribution (ii) if the numbers of columns and rows were interchanged what would be the new regenerative distribution number for this new arrangement. n (E) = 399

### References

1. Taylor, Adekola A. (2016). Derivation of formulas for position change of entities in power inductive distribution. International Journal of Scientific and Research Publications 6(7):409-420.

2. Taylor, Adekola A. (2015). Quadratic distribution patterns in Kola analysis. Mathematical Theory and Modeling 5: 60-66.

3. Taylor, Adekola A. (2015). Derivation of formulas for position change in Kola analysis. International Journal of Scientific and Research Publications 5(11):101-109.

4. Taylor, Adekola A. (2014). Logical-mathematical intelligence for teens. Mathsthoughtbook.com

5. Taylor, Adekola A. (2013). Regenerative mathematics and dimurelo puzzles for children 8-12yrs. USA: Lulu Press Inc.

6. Taylor, Adekola (2013). Card magic and my mathematical discoveries. USA:Lulu Publishing.

7. Taylor, Adekola A. (2010). Kola Analysis: An inventive approach to logical-mathematical intelligence for secondary and advanced levels. Journal of Mathematical Sciences Education 1(1): 44-53.