Blog Entries: 2

## Multiplication of numbers in a grid by vector manipulation

Consider the array of numbers on the following X-Y grid:.
X
………………………
6 11 17 24 32 41 …
3 7 12 18 25 33 …
1 4 8 13 19 26 …
0 2 5 9 14 20 …
0 1 3 6 10 15 … Y

The equation that maps this array is R(x,y) = 0.5((x+y)^2 -x +y) . Among many properties of this array, I noticed that the arithmetic sequences 1,3,5,7; 3,6,9,12; 6,10,14,18; 10,15,20,25 each form the same pattern. On further examination, this pattern in its same orientation can be sifted indefinitely on the grid and always maps an arithmetic sequence. The pattern intercepts each successive diagonal at the next number in the arithmetic series. That multiplication on this grid could be accomplished by a simple vector operation on any pattern which maps a sequence 0,1,2,3,4 …etc. turns out to be the case.
It can be shown that the vector between the point P_1 where R(A,B) = 1 and any other point (a,b) on the pattern maps each point (a',b') on the pattern into points (a",b") where R(a",b")= R(a,b) * R(a',b'). An interesting point of this analysis is that if the point (a', b’) is the same as (a,b) , then R(a",b") is a square.

As an example, consider one such pattern which forms the following table:

Point (a,b) R(a,b) Name Vector Name
of Point of vector
(15,-21) 0 P_0 <6,-7> V_0
(9,-14) 1 P_1 <0,0> V_1
(4,-8) 2 P_2 <-5,6> V_2
(0,-3) 3 P_3 <-9,11> V_3
(-3,1) 4 P_4 <-12,15> V_4
(-5,4) 5 P_5 <-14,18> V_5
(-6,6) 6 P_6 <-15,20> V_6
….

For instance the vector V_6 maps the points P_0 to P_12 as follows:

Point (a,b) (a-15,b+20) R(a-15,b+20)
P_0 (15,-21) (0,-1) 0
P_1 (9,-14) (-6,6) 6
…..
P_11 (4,1) (-11,21) 66
P_12 (9,-3) (-6,17) 72

Note how the pattern is moved by the vector such that its axis of symmetry is sifted from the N_1 diagonal (a+b = 1) to the N_6 diagonal (a+b = 6). It is easily seen that the series of R values will be an arithmetic series, and in fact multiples of the R values of the P_k (k = 1 to 12) points by 6. I believe that facturing of numbers on this grid may be done by an algorithm based on this property. Any comments.
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