# VI. SUMS and PRODUCTS: Collections 17 – 21

## 17. “Diagonal Sums” of Last Digits – in Orthogonal and Diagonal Chessboards

Last Digit SUMS are the ‘arithmetical nugget’ of ‘geometrically diagonal‘ Sums.
But there are a number of complementary concepts that need to be bridged for ‘grand unification’:

Number: Odd versus Even, ‘Player‘ versus Result;
Geometry: Vertical versus Horizontal, Orthogonal versus Diagonal,
Visualisation: Label versus Content, Single Quadrant versus Symmetrical Plane, Orthogonal versus Diagonal Chessboard.

With focus on ODD and EVEN sums of ODD and EVEN digits and numbers, there are four cases of combination:

First: ODD Last Digits as Summands in Vertical Column and in Horizontal Rows:

• ODD + ODD = EVEN

Second: ODD Last Digits in Vertical Column and EVEN Last Digits in Horizontal Rows:

• ODD + EVEN = ODD

Third: EVEN Last Digits in Vertical Column plus EVEN Digits in Horizontal Rows:

• EVEN + EVEN = EVEN

Fourth: EVEN Last Digits in Vertical Column, ODD Last Digits in Horizontal Rows:

• EVEN + ODD = ODD

## 18. “Orthogonal Products” of Last Digits in Orthogonal and Diagonal Chessboards

Geometrically, sums are diagonal to express the multitude of combinations of their summands, i.e. the ‘plurality’ of their ‘value composition’ – in a diagonal line.

Products are orthogonal because every unique combination of 2 factors produces a rectangle that is unique in its shape, e.g. 24 = 3 x 8 = 4 x 6 = 2 x 12.