Please don’t diagnose me but I have a problem. Well actually many of problems but only one under this umbrella. When I look at any tiled pavement on the floor, walls or in a graphics work, I start to make all kinds of numerical patterns. These littles bit of mental gymnastics fall into two main categories one being called the ‘what corner will I end up on’ problem. The other being the ‘how many rectangles can I make’ postulate. Both problems have some boundaries that I have imposed, but they have given rise to lovely pictures both in my mind AND the resultant solutions have equally pretty outcomes

‘what corner will I end up on’ problem. The rules here are that it applies for any rectangular field, tessellated like a chess board, with the term x being the length of one side and y the other. The problem starts a trace from one corner and moves only as a diagonal line reflecting in movement when it reaches any side until it ends up on a corner.  The only results possible are opposite corner or adjacent on the shortest side or adjacent on the longest side. The general rule for the finish point of the trace for any rectangle of size X and Y is as follows…

Group A that is when the sum of X+Y is of the set 0,4,8,12,16, 20, 24,,, or expressed as (X+Y+4)/4 is an integer

Then if X and Y are both even the outcome will be on the opposite corner.

Also if (X+3)/4 is an integer then the outcome is the X corner

Also if (X+1)/4 is an integer then the outcome is the Y corner

Group B that is when the sum of X+Y is of the set 1,5,9,13,17,21,25,,, or expressed as (X+Y+3)/4 is an integer

Group D that is when the sum of X+Y is of the set 3,7,11,15,19,23… or expressed as (X+Y+1)/4 is an integer

Then if Y is even and X is odd the outcome is the X corner

Also if X is even and Y is odd the outcome is the Y corner

Group C that is when the sum of X+Y is of the set 2,6,10,14,18,22,,, or expressed as (X+Y+2)/4 is an integer

Then the outcome will be on the opposite corner.

‘how many rectangles can I make’ postulate. The rules here are that it only works on square fields, tessellated like a chess board, with the term A being the length of one side, it is also by definition the same number of squares along the diagonal. The number n is the number of each term in the series. Each rectangle within the square starts with a diagonal line of length A and width 1, the next term is A-2 and a width of 3, following A-4 and width 5 and so on… each has an area given by the product of the length and width terms. The general result for any given set about A and with n being the consecutive terms for n being less than (A+3)/2 is (A-2n+2)(2n-1) and is known hence forth as the Craig Collect.