Ponder This Challenge - August 2011 - Triangle area / perimeter

Ponder This Challenge:

A mathematician wanted to teach his children the value of cooperation, so he told them the following:

"I chose a secret triangle for which the lengths of its sides are all integers.

To you my dear son Charlie, I am giving the triangle's perimeter. And to you, my beloved daughter Ariella, I am giving its area.

Since you are both such talented mathematicians, I'm sure that together you can find the lengths of the triangle's sides."

Instead of working together, Charlie and Ariella had the following conversation after their father gave each of them the information he promised.

Charlie: "Alas, I cannot deduce the lengths of the sides from my knowledge of the perimeter."

Ariella: "I do not know the perimeter, but I cannot deduce the lengths of the sides from just knowing the area. Maybe our father is right and we should cooperate after all."

Charlie: "Oh no, no need. Now I know the lengths of the sides."

Ariella: "Well, now I know them as well."

For this month's challenge, find the lengths of the triangle's sides and explain the dialogue above.

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Solution

• Click here to view the solution

  The three solutions we found are (4,4,3); (4,4,4); and (4,4,6).

  Here's the explanation based on the dialog for the first example:

  Charlie knows the perimeter 11, but he cannot know the length of the sides, since (5,5,1); (5,4,2); and (5,3,3) are legitimate triangles, all having the same perimeter 11.

  Ariella know the area 3/4*sqrt(55), but cannot distinguish between (4,4,3) and (7,6,2).

  Charlie checks all four possibilities stated above and sees that in all cases, except (4,4,3), the area determines the triangle's sides uniquely. Therefore, he can deduce that the triangle has to be (4,4,3).

  Ariella reasons that if the triangle's sides were (7,6,2) Charlie would not have been able to find the sides since not only (7,6,2) but also (7,5,3) could have led to the same dialog. Therefore she can deduce the sides.

  Note that (2,2,3) and (1,4,4) are common mistakes. It explains the first three statements, but not the last one -- that Ariella cannot tell because Charlie could have known in both cases.

  John P. Robertson proved that the three triangles above are the only possible solutions by giving formulas to construct special triangles for large enough perimeters and checking all the possibilities for smaller perimeters.

  There are some variants, such as when the triangle area is an integer too (Heronian triangle); or when we know that the triangle is not isosceles; or replacing the perimeter with the circumference of the bounding circle; etc.

Solvers

• Lukasz Bolikowski (07/30/2011 12:47 PM EDT)
• Vladimir Sedach (07/30/2011 02:29 PM EDT)
• Mark Mixer (07/30/2011 10:24 PM EDT)
• Hakan Summakoglu (07/31/2011 08:07 AM EDT)
• Qinsheng Lai (07/31/2011 10:23 PM EDT)
• William Heller (07/31/2011 10:44 PM EDT)
• Lewei Weng (08/01/2011 08:47 AM EDT)
• Andrew D Kidd (08/01/2011 01:48 PM EDT)
• Lael & Adam Daire (08/01/2011 02:26 PM EDT)
• Kurt Hectic (08/01/2011 03:37 PM EDT)
• Joseph DeVincentis (08/01/2011 04:19 PM EDT)
• Ryan Manuel (08/01/2011 04:27 PM EDT)
• Jan Fricke (08/01/2011 05:07 PM EDT)
• Joe Fendel (08/01/2011 08:33 PM EDT)
• Alan Murray (08/02/2011 12:37 AM EDT)
• Liubing Yu (08/02/2011 01:57 AM EDT)
• Michael Brand (08/02/2011 03:24 AM EDT)
• Mathias Schenker (08/02/2011 04:16 AM EDT)
• Blatter Christian (08/02/2011 08:28 AM EDT)
• Gideon Kojokaro (08/02/2011 10:19 AM EDT)
• Edward Kim (08/02/2011 01:30 PM EDT)
• Christopher Marks (08/02/2011 02:04 PM EDT)
• Shilei Zang (08/02/2011 03:48 PM EDT)
• Donald T Dodson (08/02/2011 03:55 PM EDT)
• Christian Matt (08/02/2011 08:39 PM EDT)
• Leif Jensen (08/02/2011 10:50 PM EDT)
• Lu Wang (08/03/2011 03:41 AM EDT)
• Jan Braunisch (08/03/2011 05:19 AM EDT)
• Puvichakravarthy Ramachandran (08/03/2011 05:58 AM EDT)
• Jane Qian (08/03/2011 08:07 AM EDT)
• Dan Dima (08/03/2011 07:48 AM EDT)
• Danny Han (08/03/2011 06:52 PM EDT)
• Ian Smith (08/04/2011 01:41 AM EDT)
• Nate Lokers (08/04/2011 03:05 AM EDT)
• Frederik Kaster (08/04/2011 04:41 AM EDT)
• Wu ChengYuan (08/04/2011 02:12 PM EDT)
• Lukáš Polácek (08/04/2011 05:01 PM EDT)
• Jeff Spirko (08/04/2011 05:15 PM EDT)
• Viktor (08/04/2011 07:47 PM EDT)
• Radu Borza (08/04/2011 11:18 PM EDT)
• Grzegorz Materna (08/05/2011 08:20 AM EDT)
• Bethany Emerson (08/05/2011 03:42 PM EDT)
• Albert Stadler (08/06/2011 06:35 PM EDT)
• Jeff Steele (08/06/2011 08:17 PM EDT)
• Vikranth Inti (08/06/2011 11:43 PM EDT)
• Shyam M (08/07/2011 02:27 PM EDT)
• N (08/07/2011 10:28 PM EDT)
• oah Easterly (08/07/2011 10:28 PM EDT)
• Fletcher Dostie (08/08/2011 04:05 PM EDT)
• Thomas Rohr (08/09/2011 12:37 PM EDT)
• Athena Health Problem Solving Group (08/09/2011 02:25 PM EDT)
• Matt Dobrin (08/09/2011 05:16 PM EDT)
• Clive Tong (08/09/2011 07:16 PM EDT)
• Prithu Tiwari (08/10/2011 09:28 AM EDT)
• Aditya Kamat (08/10/2011 10:14 AM EDT)
• Balazs Ugron (08/10/2011 07:46 PM EDT)
• John Tromp (08/11/2011 11:58 AM EDT)
• Giovanni E.Pazienza (08/11/2011 07:11 PM EDT)
• Harinarayanan E V (08/13/2011 10:40 AM EDT)
• Daniel Bitin (08/13/2011 04:07 PM EDT)
• Srinivasan (08/14/2011 01:00 PM EDT)
• Gal Dor (08/14/2011 11:50 PM EDT)
• Ganesh Lakshminarayana (08/15/2011 06:39 PM EDT)
• Werner Gillijns (08/15/2011 06:21 PM EDT)
• Pritibhushan Sinha (08/16/2011 01:11 AM EDT)
• Guler Amos (08/17/2011 01:16 PM EDT)
• Thomas Mack (08/20/2011 07:58 PM EDT)
• Peter Gerritson (08/21/2011 02:16 AM EDT)
• Traian Delca (08/21/2011 12:00 PM EDT)
• John P. Robertson (08/23/2011 12:06 AM EDT)
• Fabio Filatrella & Marco Quargentan (08/23/2011 11:24 AM EDT)
• Radu-Alexandru Todor (08/24/2011 05:14 AM EDT)
• J. Noble Pendergrass (08/24/2011 02:32 PM EDT)
• Anoop Ghanwani (08/24/2011 09:25 PM EDT)
• Umut Uludag (08/26/2011 01:01 PM EDT)
• Chris Shannon (08/26/2011 04:00 PM EDT)
• Georg Winkler (08/26/2011 06:07 AM EDT)
• Máté Bartalos (08/27/2011 10:39 AM EDT)
• Mojtaba Tefagh (08/27/2011 04:50 PM EDT)
• Maris Van Sprang (08/28/2011 04:37 PM EDT)
• Praveen Vaka (08/28/2011 06:30 PM EDT)
• Robert Buckley (08/29/2011 12:46 PM EDT)
• Rubén Hernández Aza (08/30/2011 05:56 PM EDT)
• Andrea Andenna (08/31/2011 04:52 AM EDT)
• Piet Orye (08/31/2011 05:51 AM EDT)
• Amos Shapir (08/31/2011 12:16 PM EDT)
• Vignesh Sethuraman (08/31/2011 04:27 PM EDT)
• Ilya Khivrich (08/31/2011 08:13 PM EDT)
• Manuel Sanchez-Gestido (08/31/2011 08:25 PM EDT)
• BJOA Puzzle Team (08/31/2011 10:04 PM EDT)