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# File project-euler/147/euler-147-analysis.txt

` `
` C(n,step) = n*2-3`
` `
`+General diagonal rectangles analysis:`
`+-------------------------------------`
` `
`+Since the rectangles stop at midpoint, we can treat the board as having`
`+units of 2. The points can be either (2x, 2y) or (2x+1, 2y+1). If the origin`
`+is point (Ox,Oy), then the dimensions of a w*h diagonal rectangle are:`
`+`
`+( Ox , Oy )`
`+( Ox - w , Oy + w )`
`+( Ox - w + h , Oy + w + h )`
`+( Ox + h , Oy + h )`
`+`
`+The complete span is (w+h) * (w+h) (but the origin of the span is not at`
`+a lattice point).`
`+`
`+So we need the straight square with dimensions x==[-w, h] * y==[0, w+h]`
`+`
`+Without loss of generality, let's assume that w >= h.`
`+`
`+For b[x] = 2x' ; b[y] = 2y' (board dimensions), the points for such a span`
`+are:`
`+`
`+1. For even points:   x==[w , 2x' - h] * y==[0, 2y'-(w+h)]`
`+(divided by 2 and rounded down and while ignoring values beyond the board`
`+dimensions.)`
`+`
`+2. For the +(1,1) points: there are (x'-1) * (y'-1) points like that.`
`+    and they follow the same rules:`
`+    (1, 3, 5...) * (1, 3, 5...)`