Prime partitions of a rectangle
ABCD is a rectangle such that AB ≥ AD. AB, AD, AE and AF are integer lengths. A configuration is called “first” when the 8 surfaces Si (i = 1, 2, ..., 8) are integer and prime among themselves (globally and not two by two). Two prime configurations G and G' are considered equivalent if: {S'1, S'2, S'3, S'4, S'5, S'6, S'7, S'8} = {S1, S2, S3, S4, S5, S6, S7, S8} (we are talking about sets). Among a set of equivalent primary configurations, it is agreed to call the main configuration the one where AB is minimal (which amounts to choosing the one with minimal perimeter or the one closest to a square...).
Examples:
(AB, AD, AE, AF) = (10, 6, 5, 3) is the smallest main configuration, with: (S1, S2, S3, S4, S5, S6, S7, S8) = (10, 3 , 3, 16, 12, 12, 2, 2), and a total area of 60.
(21, 16, 14, 8) is a main configuration which gives (70, 8, 21, 99, 48, 63, 6, 21) and a total area of 336.
Let N be the number of principal principal configurations when AB ≤ 100. These N configurations only give (N – k) distinct total surfaces, because k of them are obtained twice. Let P be the sum of these k areas.
What is the value of N x P?
I first looked for relationships between the different surfaces of the triangles and I found that:
S6 + S7 = S5 + S8
S4 = S1 + S2 + S3
I have not found a use for these relationships to optimize computer problem solving. so I wrote a program to solve this problem by brute force. My program finds 353 integer configurations. Among these 353 whole configurations I find 29 prime configurations:
(10, 6, 5, 3): (10, 3, 3, 16, 12, 12, 2, 2) Area: 60
(21, 16, 14, 8): (70, 8, 21, 99, 48, 63, 6, 21) Area: 336
(21, 20, 7, 10): (49, 35, 15, 99, 105, 90, 21, 6) Area: 420
(24, 14, 16, 7): (70, 8, 21, 99, 48, 63, 6, 21) Area: 336
(28, 15, 14, 5): (49, 15, 35, 99, 90, 105, 6, 21) Area: 420
(30, 28, 20, 21): (245, 28, 15, 288, 112, 90, 42, 20) Area: 840
(40, 21, 30, 14): (245, 15, 28, 288, 90, 112, 20, 42) Area: 840
(40, 24, 16, 20): (184, 96, 5, 285, 192, 75, 120, 3) Area: 960
(42, 8, 28, 4): (70, 8, 21, 99, 48, 63, 6, 21) Area: 336
(42, 10, 14, 5): (49, 35, 15, 99, 105, 90, 21, 6) Area: 420
(42, 40, 36, 20): (405, 8, 126, 539, 112, 294, 7, 189) Area: 1680
(48, 20, 40, 8): (184, 5, 96, 285, 75, 192, 3, 120) Area: 960
(56, 15, 42, 10): (245, 15, 28, 288, 90, 112, 20, 42) Area: 840
(56, 30, 48, 15): (405, 8, 126, 539, 112, 294, 7, 189) Area: 1680
(60, 52, 24, 26): (429, 216, 130, 775, 720, 650, 135, 65) Area: 3120
(65, 48, 26, 24): (429, 216, 130, 775, 720, 650, 135, 65) Area: 3120
(66, 10, 55, 6): (187, 5, 33, 225, 50, 99, 6, 55) Area: 660
(70, 6, 35, 2): (49, 15, 35, 99, 90, 105, 6, 21) Area: 420
(70, 24, 60, 12): (405, 8, 126, 539, 112, 294, 7, 189) Area: 1680
(77, 72, 33, 42): (913, 396, 175, 1484, 1188, 980, 308, 100) Area: 5544
(80, 12, 32, 10): (184, 96, 5, 285, 192, 75, 120, 3) Area: 960
(80, 33, 24, 22): (341, 294, 40, 675, 630, 400, 245, 15) Area: 2640
(80, 77, 56, 66): (2046, 189, 40, 2275, 735, 400, 405, 70) Area: 6160
(84, 4, 56, 2): (70, 8, 21, 99, 48, 63, 6, 21) Area: 336
(84, 20, 72, 10): (405, 8, 126, 539, 112, 294, 7, 189) Area: 1680
(90, 24, 54, 4): (148, 27, 300, 475, 405, 600, 5, 200) Area: 2160
(95, 36, 57, 24): (836, 144, 95, 1075, 540, 475, 160, 95) Area: 3420
(96, 10, 80, 4): (184, 5, 96, 285, 75, 192, 3, 120) Area: 960
(100, 72, 50, 16): (575, 180, 784, 1539, 1620, 2016, 45, 441) Area: 7200
The values are inconsistent with the problem or I didn't understand everything