From Al Zimmermann:
M=1: (4,4), (3,6) M=2: (8,8), (6,12), (5,20) M=3: (12,12), (10,15), (9,18), (8,24), (7,42) M=4: (16,16), (12,24), (10,40), (9,72) M=5: (20,20), (15,30), (14,35), (12,60), (11,110) Some of these rectangles are just "multiples" of smaller rectangles. Specifically, if M = Area(x,y) / Perimeter(x,y) then kM = Area(kx,ky) / Perimeter(kx,ky). I think such rectangles are inherently less interesting. The abbreviated list of rectangles which are not multiples of smaller ones (that is, where M, x, and y have no common divisor other than 1) is: M=1: (4,4), (3,6) M=2: (5,20) M=3: (10,15), (8,24), (7,42) M=4: (9,72) M=5: (14,35), (12,60), (11,110) Extending this list of "primitive" rectangles to M=10 yields: M=6: (21,28), (13,156) M=7: (18,63), (16,112), (15,210) M=8: (17,272) M=9: (22,99), (20,180), (19,342) M=10: (36,45), (21,420) The largest rectangle for a given M appears to always be (2M+1,4M^2+2M). [KD: Stan Morrice stated this as "the dimensions are X=2*M+1, and Y=2*M*X." I thought that was a nice compact statement.] (2M+2,2M^2+2M) and (2M+4,M^2+2M) also work, although neither is primitive if M is even. > Repeat for perimeter equal to M times the area. M=1: (4,4), (3,6) M=2: (2,2) M=3: (1,2) M=4: (1,1) M=5:From Ravi Subramanian:
Let the dimensions of the rectangle be a, b. Area = a * b Perimeter = 2 * (a + b) First part : Area = M * Perimeter a * b = M * 2 * (a + b) 1 / a + 1 / b = 1 / (2 * M) let (2 * M) = N 1 / a + 1 / b = 1 / N Let a <= b. Hence N < a <= 2 * N. Let a = (N + d) with 0 < d <= N So 1 / (N + d) + 1 / b = 1 / N so b = N * (N + d) / d => d is a factor N^2 => d is a product of 2 factors of N. So considering various M's M N d a b P A 1 2 1 3 6 18 18 2 4 4 16 16 2 4 1 5 20 50 100 2 6 12 36 72 4 8 8 32 64 3 6 1 7 42 98 294 2 8 24 64 192 3 9 18 54 162 4 10 15 50 150 6 12 12 48 144 4 8 1 9 72 162 648 2 10 40 100 400 4 12 24 72 288 8 16 16 64 256 5 10 1 11 110 242 1210 2 12 60 144 720 4 14 35 98 490 5 15 30 90 450 10 20 20 80 400 Second part : Perimeter = M * Area 2 * (a + b) = M * a * b 1 / a + 1 / b = M / 2 Let a <= b. So 1 / a >= M / 4 a <= 4 / M and a > 2 / M No solution for M >= 5! M a b A P 1 3 6 18 18 4 4 16 16 2 2 2 4 8 3 1 2 2 6 4 1 1 1 4