Let me know whether I did understand your question.

[I add comments in red and between [ and ]]

[If I understand you well, you are interested in the relation between the number n, its square, the cross addition of the square and the base number. You have found a number of instances where the latter can be derived directly from the cross addition of the square. To explore the situation I have constructed an Excel demonstration. It is attached. The table shows an interesting pattern for the cross addition of n2 (in colour). But this is not what you appear to be after. Note that column F indicates the results of various ‘tricks’ to compute the base number from the cross additions of the square. Just exercises.

It seems useful to consider a general form of the problem. Let me start by considering the number n.

If we are in between 0 and 10, the number n can be written as x.

If we are in between 10 and 20, the number n can be written as xy, or 10.x + y.

If we are in between 20 and 29, the number n can be written as xy, or 210.x +y.

If we are in between 100 and 109, the number n can be written as xyz, or 100.x+10.y+z (where y = 0).

Etc.

I think your problem is to solve the following equation (for numbers between 10 and 20):

(10.x+y)^{2}+ = (100x^{2}+20.x.y+y^{2})+ = 2.(10.x+y)-1.

It appears that there is no general solution. It is possible to find solutions by looking for combinations, for example.

If n=19, the equation is 100.1+20.9+81=361. The base number is 37, which one can derive by the trick of taking 30+(61)+=37.

If n=20, the equation is 100.4+20.0+02=400. The base number is 39, but 40+(00)+=40. Considering 40 separately is a trick that is not part of the mapping.

If n=21, the equation is 100.4+20.2.1+1=441. The base number is 41, but 40+4+1=45.

Etc.

It would seem that the number of numbers for which the base number is the same as (10x+y)2+ is quite limited, and also not densely or equally distributed over the set of natural numbers.

Did I understand your problem?]

On 21/02/2014 09:59, Graham Freestone wrote:

Dear Gerard

Entirely aside from the forum I have a problem which you might be able to understand. These are sections taken from some online notes I have written but I think the issue is clear. Any thoughts appreciated.

Best

Graham

** **

**The Numerical system and the reduction that squaring and cross addition performs upon it.**

The foundation of the system is derived from the following squaring of base 10 coupled with the classic numerological process of cross addition formalized as n+ e.g.

(13)+=(1+3).

The rule was once observed that ‘the highest incarnation of a number is itself multiplied by itself **(ignore this kind of language in this instance)**‘. The accretion of this rule was taken to heart and thus derives results even though the accretion itself is of course contingent and the term ‘highest incarnation’ subject to incoherentism. (Refer to the **Tractatus** for the understanding of these terms).

When these rules are applied the number system shows itself to be made up of these numbers only:

1=1

2=4=(16)+=7=(49)+=(13)+=4 and will repeat this endlessly thus we can write 4v7

3=9=(81)+=9

4=4v7

5=(25)+=4v7

6=(36)+=9

7=4v7

8=(64)+=(10)+=1

9=9

Thus from this perspective, taking this rule as truth, the highest manifestation of the numbers is 1, and oscillation of 4v7 and 9.

The wary observer will be aware that all of this is contingent upon base 10 and can represent nothing greater than this one numerical perspective. What must be understood in the light of the Tractatus is that the accretion of base 10 is very powerful, and thus whilst contingent in one sense, in another is the only thing worth working with to generate further connections with this world.

[I understand the above to mean that every natural number can be mapped onto the set of single numbers 1, 4, 7 and 9. The mapping is the result of the operation of cross addition. As mentioned, there is a pattern: see the attachment, the coloured numbers.]

**Cross addition and the problem itself.**

Cross addition ((123)+=6) suggests the possibility of a relation, indeed says that under certain condition (a particular base) there is a relation between a number higher than the highest single integer and one of the single integers. The general impression would be that this is in a sense arbitrary or at least meaningless. What has 53 e.g. got to do with 8? Very little other than the cross addition relationship.

What is truly fascinating is the that the squares of 4 and 7 give numbers whose cross addition then has a demonstrable reality in a triangle comprised of units 1 at the top 3 on the next line down, then 5, 7 and so on.

Triangles of this kind of unit construction are remarkable as they also provide squares. The number that will be squared is the height of the triangle. So if I have a triangle of 2 height, the total number of units in the triangle will be 4, if 3 it will be nine and so on.

The relation to the issue of cross addition is as follows. If I have a triangle of a height of 4, necessarily it will be comprised of 16 units. The base however will be *7 units*(1+6). There is one relation uncovered here for the base will always have the relation to the height 2n-1. If the height is 12 the base is 23 and so on.

The second relation that is more interesting to us here is the one concerning squares. Squares of 4s and 7s even of cross addition ones will always reduce to 7s or 4s respectively but the base seems to often (though not always) reveal a relation between the number itself, the square and the cross addition of the square.

**These are the most concrete examples:**

4 becomes 16 becomes 7 (the base units of 4)

7 becomes 49 becomes 13 (the base units of the 7 triangle) becomes 4

**These require a tweak to make them work but are still quite convincing.**

13 becomes 169 becomes (curiously by preserving the first two digits as a whole number) 16+9=25 is the base number.

16 becomes 256 becomes 31 (by the same logic above) which is the base number and also reduces to 4.

22 becomes 484 and a similar logic derives the base. This time we extract 40 and add 8+4=12=3, re-add them and we have 43, the base number.

31 gives us 961, if we cross out the 9 for 9=0 in base 10 cross addition we immediately have the base number again.

**This one has no link though.**

25*25=625=13=4, or course the 7/4 transformation is preserved but the base relation is not. The base would be 49 and 625 does not have a relation to it. And no doubt there are others.

**The Crux**

Given that the relation between 4 and 13 contingent entirely on the base we write it in, how is it that it seems that this relation is *necessary* even if only in some cases? I am not saying anything mystical here, the mystical can be bolted onto it, but that’s not the point? Isn’t it possible there is something numerically interesting here? ( I did not say mathematical ;-))