sum_of_two_rectangles_solutions.sf

#!/usr/bin/ruby

# Author: Trizen
# Date: 18 March 2022
# https://github.com/trizen

# Represent an integer as a sum of two rectangles:
#
#   n = a*b + c*d
#
# with a,b,c,d > 1 and a != b and c != d.

func difference_of_two_squares_solutions(n) {

    n.divisors.map {|d|

        break if (d*d >= n)

        var a = d
        var b = n/d

        (a+b).is_even || next

        var x = (a + b)/2
        var y = (b - a)/2

        [x, y]
    }
}

func sum_of_two_rectangles_solutions(n) {

    var solutions = []

    for x in (1 .. (n>>1)) {

        var y = (n - x)

        var ab = []
        var cd = []

        difference_of_two_squares_solutions(x).each_2d {|a,b|
            ab << [a+b, a-b] if (a-b > 1)
        }

        difference_of_two_squares_solutions(y).each_2d {|c,d|
            cd << [c+d, c-d] if (c-d > 1)
        }

        [ab,cd].cartesian {|x,y|
            solutions << [x..., y...]
        }
    }

    return solutions.sort
}

var n = 48

sum_of_two_rectangles_solutions(n).each_2d {|a,b,c,d|
    say "#{a*b + c*d} = #{a}*#{b} + #{c}*#{d}"
}

__END__
48 = 4*2 + 10*4
48 = 4*2 + 20*2
48 = 5*3 + 11*3
48 = 6*2 + 18*2
48 = 6*4 + 6*4
48 = 6*4 + 12*2
48 = 7*3 + 9*3
48 = 8*2 + 8*4
48 = 8*2 + 16*2
48 = 10*2 + 14*2
48 = 12*2 + 6*4
48 = 12*2 + 12*2