September 2024 - Challenge
This riddle was suggested by Latchezar Christov - thanks Latchezar!
We will call two triangles "sibling pairs" if they are two non-identical triangles with integer side lengths respectively (a,b,c_1) and (a,b,c_2), where c_1 and c_2 are the only side of each triangle with a different length from the other triangle, since there are two common side lengths a,b **and** both triangles must have a common nonzero area.
An example is the pair of triangles with side lengths (5,5,6) and (5,5,8). The area of both is 12. Another example is (7,4,7) and (7,4,9) with area \sqrt{180}.
For some a and b, there may not be any sibling pair at all (e.g., a=3 and b=2) and for some, there may be more than one. For example, if a=24 and b=23, three sibling pairs exist with the following side lengths:
- (24,23,19) and (24,23,43), area = \sqrt{41580}
- (24,23,23) and (24,23,41), area = \sqrt{55440}
- (24,23,29) and (24,23,37), area = \sqrt{71820}
Your goal: Find two integers a,b such that a>b and there are **exactly** 50 sibling pairs with common side lengths a and b.
**A bonus** "*" will be given for solving the above problem with the additional constraint that at least two of those 50 sibling pairs must have **integer** areas. The areas of the remaining pairs may be irrational.
We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!
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Challenge:
29/08/2024 @ 12:45 PM EST
Solution:
05/10/2024 @ 12:00 AM EST
List Updated:
12/09/2024 @ 11:30 AM EST
Lazar Ilic(29/8/2024 3:42 PM IDT)
*King Pig(30/8/2024 2:17 AM IDT)
Daniel Chong Jyh Tar(30/8/2024 2:26 AM IDT)
*Bertram Felgenhauer(30/8/2024 3:44 AM IDT)
Alper Halbutogullari(30/8/2024 5:03 AM IDT)
Lorenz Reichel(30/8/2024 1:27 PM IDT)
Alex Fleischer(30/8/2024 3:34 PM IDT)
*Sanandan Swaminathan(30/8/2024 8:32 PM IDT)
John Tromp(30/8/2024 10:07 PM IDT)
*Bert Dobbelaere(31/8/2024 12:55 AM IDT)
Julian Ma(31/8/2024 4:03 AM IDT)
Dan Dima(31/8/2024 9:17 AM IDT)
Loukas Sidiropoulos(31/8/2024 11:21 AM IDT)
Yi Jiang(31/8/2024 1:36 PM IDT)
Viplov Jain(31/8/2024 4:27 PM IDT)
Paul Lupascu(31/8/2024 5:39 PM IDT)
John Spurgeon(31/8/2024 7:21 PM IDT)
*Victor Chang(31/8/2024 8:09 PM IDT)
Kipp Johnson(1/9/2024 3:02 PM IDT)
Harald Bögeholz(1/9/2024 3:16 PM IDT)
Gary M. Gerken(1/9/2024 5:15 PM IDT)
*Yan-Wu He(1/9/2024 6:56 PM IDT)
Hugo Pfoertner(2/9/2024 12:35 AM IDT)
Marco Bellocchi(2/9/2024 1:41 AM IDT)
Shirish Chinchalkar(2/9/2024 2:32 AM IDT)
Jawher Jandoubi(2/9/2024 2:42 AM IDT)
Richard Gosiorovsky(2/9/2024 9:14 AM IDT)
Andreas Knüpfer(2/9/2024 9:28 AM IDT)
Jiri Spitalsky(2/9/2024 11:34 AM IDT)
*Guangxi Liu(2/9/2024 1:10 PM IDT)
Reiner Martin(2/9/2024 2:37 PM IDT)
Dan Ismailescu(2/9/2024 6:54 PM IDT)
Daniel Bitin(2/9/2024 11:49 PM IDT)
Michael Liepelt(3/9/2024 8:27 AM IDT)
*Frederick Vu(3/9/2024 11:40 AM IDT)
Sri Mallikarjun J(3/9/2024 4:20 PM IDT)
Juergen Koehl(3/9/2024 9:55 PM IDT)
Guglielmo Sanchini(3/9/2024 10:08 PM IDT)
Robin Guilliou(4/9/2024 4:03 PM IDT)
Arthur Vause(4/9/2024 7:26 PM IDT)
*Lawrence Au(4/9/2024 8:06 PM IDT)
Janko Sustersic(4/9/2024 8:40 PM IDT)
Adrian Neacsu(5/9/2024 1:07 AM IDT)
Fabio Michele Negroni(5/9/2024 1:57 PM IDT)
Piyush Agrawal(5/9/2024 3:23 PM IDT)
Jesse Rearick(6/9/2024 9:22 AM IDT)
Thomas Egense(6/9/2024 10:07 AM IDT)
Vladimir Volevich(7/9/2024 12:24 AM IDT)
*Andrea Andenna(7/9/2024 12:34 AM IDT)
*Mark Beyleveld(7/9/2024 4:40 PM IDT)
Alex Izvalov(7/9/2024 5:30 PM IDT)
Reda Kebbaj(8/9/2024 4:33 AM IDT)
*Radu-Alexandru Todor(8/9/2024 2:06 PM IDT)
Maksym Voznyy(9/9/2024 1:54 AM IDT)
Ashfaque Shaikh(9/9/2024 5:54 PM IDT)
Amos Guler(9/9/2024 9:45 PM IDT)
*Martin Thorne(9/9/2024 9:49 PM IDT)
Gyozo Nagy(9/9/2024 10:08 PM IDT)
Jared Kittleson(9/9/2024 10:16 PM IDT)
Karl-Heinz Hofmann(10/9/2024 11:09 AM IDT)
Mathias Schenker(10/9/2024 1:53 PM IDT)
Joaquim Carrapa(11/9/2024 1:30 AM IDT)
Karl D'Souza(11/9/2024 2:40 AM IDT)