- Find the dimensions of all triangles that have area equal numerically to M times the perimeter, where all sides are whole numbers. (M=1,2,3,4,5)
- Repeat for perimeter equal to M times the area.

Source: Extended from a puzzle submitted by Ravi Subramanian (where M=1).

Solutions were received from Philippe Fondanaiche, Le Ba Nguyen, Ravi Subramanian, Kirk Bresniker, Al Zimmermann, and Sandy Thompson. The smallest-perimeter solutions are:

M=1: (6, 8, 10); Perimeter = 24, Area = 24 M=2: (13, 14, 15); Perimeter = 42, Area = 84 M=3: (20, 20, 24); Perimeter = 64, Area = 192 M=4: (26, 28, 30); Perimeter = 84, Area = 336 M=5: (30, 39, 39); Perimeter = 108, Area = 540 M=1/2: (3, 4, 5); Area = 6, Perimeter = 12 [This is the only solution.] M=1/3: None. M=1/4: None. M=1/5: None.Every solver used Heron's formula:

- s = (a+b+c)/2
- A = sqrt[s(s-a)(s-b)(s-c)]

Mail to Ken

Update 11/15/99: Oliver Stone solved this puzzle in a very thorough way, so I thought I'd include his solution. It's a Word 97 file: Oliver's Stone's Solution.

Al Zimmermann's exhaustive list of primitive solutions follows:

I've only attempted the first half of the problem, in which we seek triangles whose areas are numerically equal to m times their perimeters. As in the previous rectangle problem, if a triagle (a,b,c) is a solution for m, then (ka,kb,kc) is a solution for km. Call a solution "primitive" if it cannot be so dervived from a solution for a smaller m (that is, if the greatest common divisor of (a,b,c,m) is 1). At the bottom of this is a list of all primitive solutions for m = 1 thru 5. The following formulae work: (4m+1, 8m^2+4m, 8m^2+4m+1) [always a right triangle] (8m^2+1, 8m^2+2, 16m^2+1) (4m^2+2, 4m^2+4, 8m^2+2) [not primitive if m is even] The very last triangle for a given m seems always to be: (4m^2+2, 16m^4+8m^2+1, 16m^4+12m^2+1) And then there's these: (4m^2+3, 8m^4+6m^2+1, 8m^4+10m^2+2) (4m^2+5, 4m^4+5m^2+1, 4m^4+9m^2+4) Note that some solutions, such as (9,10,17) for m=1, can be derived from more than one formula. Specific solutions: 1: (6,8,10) 1: (5,12,13) 1: (9,10,17) 1: (7,15,20) 1: (6,25,29) 2: (13,14,15) 2: (15,15,24) 2: (11,25,30) 2: (15,26,37) 2: (10,35,39) 2: (9,40,41) 2: (33,34,65) 2: (25,51,74) 2: (9,75,78) 2: (11,90,97) 2: (21,85,104) 2: (19,153,170) 2: (18,289,305) 3: (20,20,24) 3: (17,25,26) 3: (17,25,28) 3: (20,21,29) 3: (16,30,34) 3: (15,34,35) 3: (22,26,40) 3: (14,48,50) 3: (25,33,52) 3: (24,35,53) 3: (16,52,60) 3: (20,51,65) 3: (14,61,65) 3: (19,60,73) 3: (38,40,74) 3: (35,44,75) 3: (32,50,78) 3: (30,56,82) 3: (29,60,85) 3: (13,84,85) 3: (28,65,89) 3: (26,80,102) 3: (25,92,113) 3: (17,105,116) 3: (13,122,125) 3: (24,110,130) 3: (14,130,136) 3: (73,74,145) 3: (23,140,159) 3: (55,111,164) 3: (49,148,195) 3: (16,195,205) 3: (22,200,218) 3: (46,185,229) 3: (43,259,300) 3: (21,380,397) 3: (41,370,409) 3: (40,481,519) 3: (39,703,740) 3: (38,1369,1405) 4: (25,29,36) 4: (25,38,51) 4: (21,89,100) 4: (25,84,101) 4: (35,73,102) 4: (51,98,145) 4: (17,144,145) 4: (27,219,240) 4: (129,130,257) 4: (117,145,260) 4: (105,169,272) 4: (97,195,290) 4: (91,225,314) 4: (21,340,353) 4: (85,273,356) 4: (81,325,404) 4: (78,385,461) 4: (75,481,554) 4: (17,550,555) 4: (73,585,656) 4: (25,803,822) 4: (70,897,965) 4: (35,1122,1153) 4: (18,1157,1165) 4: (69,1105,1172) 4: (67,2145,2210) 4: (66,4225,4289) 5: (30,39,39) 5: (28,45,53) 5: (26,51,55) 5: (33,41,58) 5: (24,70,74) 5: (36,51,75) 5: (34,56,78) 5: (31,68,87) 5: (54,58,104) 5: (22,120,122) 5: (23,123,130) 5: (26,136,150) 5: (53,109,156) 5: (29,150,169) 5: (34,174,200) 5: (102,104,202) 5: (92,117,205) 5: (21,220,221) 5: (78,152,226) 5: (77,156,229) 5: (72,182,250) 5: (43,218,255) 5: (27,275,292) 5: (65,252,313) 5: (24,306,318) 5: (62,312,370) 5: (201,202,401) 5: (60,377,433) 5: (151,303,452) 5: (22,447,455) 5: (21,533,538) 5: (26,525,541) 5: (57,572,625) 5: (126,505,629) 5: (121,606,725) 5: (56,702,754) 5: (30,754,776) 5: (38,763,795) 5: (23,1156,1167) 5: (111,1111,1220) 5: (54,1352,1402) 5: (37,1853,1884) 5: (106,2121,2225) 5: (53,2652,2701) 5: (105,2626,2729) 5: (103,5151,5252) 5: (102,10201,10301)