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The tropical shrimp, Alpheus heterochaelis (also called ‘pistol shrimp’), has two claws, one resembling an oversize boxing glove, which it uses to stun and kill prey, such as small crabs, by snapping the oversize claw shut.
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Previous Puzzles of the Month + Solutions

August-September 2007, Puzzle nr 113
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Puzzle # 113  Italiano italiano Français francais
Difficulty level: bulbbulbbulb, basic geometry knowledge.

Indecisive triangle
   If ED = 23 units, and the value of the sides of the green square ABCD is a multiple of 11, what is the area of the red triangle AFE? Hint: the length value of EF is not irrational. Find the very shortest way to solve this puzzle and use only basic geometry, trigonometry is not allowed!

puzzle 113
(the drawing is only representative, it is not scaled!)

You can solve this puzzle by using only visual thinking skills and logic...

• Since ABCD is a square and n = AD = AB.
• Then, triangles ADF and BDF are of the same area (DF x n/2).
• Triangles EDF and BDF cover the surface of triangle EDB.
• Thus, the area A of triangle AFE = the area A of triangle EDB, which is m x n/2 (see visual proofs below).

puzzle solution 113


puzzle 113 solution 2

We can also prove that the area A of triangle AFE equals m x n/2 with the following equation:
AFE = ABE - ABF = (AB x AE)/2 - (AB x AD)/2 =
= (AB/2)(AE-AD) = (AB/2)(DE) = m x n/2 = 23n/2

To get a numerical solution, we have now to find a suitable value for n (or AB).

Triangles ABE and EDF are similar. Ratios for similar triangles:
m/n = AE/DE = BE/EF

AE, DE, and EF all have rational lengths, so BE must also be rational. Since AE and AB are integers, BE can be rational only if it is also integer.

The only right angled triangles with integer side lengths are called Pythagorean triangles, and their respective side-length values, Pythagorean triples. Therefore, we have to look for Pythagorean triples of the form: 11y, 11y+23, z

The triple 33, 56, 65 is found easily.

The value of n (or AB) being 33 units, the area of triangle AFE is:
23 x 33/2 = 379.5 square units

amaresh solutionAmaresh G. S. has sent us another visual solution that proves the area A of triangle AFE equals m x n/2.

In rectangle ABGE opposite, EB is a diagonal, and line HI passes through F and is perpendicular to AB.

Area A of rectangle AIFD = Area A of rectangle HFCG.

Area A of triangle AFD is half the area A of rectangle AIFD.

Area A of triangle FCG is half the area A of rectangle HFCG.

Therefore, area A of triangle AFD = area A of triangle FCG.

DFE + FCG = half the area A of rectangle DCGE, i.e. m x n/2.


cup winnerThe Winners of the Puzzle of the Month are:
Larry Bickford, USA flag usa
Amaresh G.S., India indian flag



© 2003 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!

More Math Facts behind the puzzle

exampleIt is possible to generate Pythagorean triples by supplying two different positive integer values for m and n in the diagram opposite. You can multiply out these terms and check that:

(m2n2)2 + (2mn)2 = (m2 + n2)2

Once one triple has been found, you can generate many others by just scaling up all the sides by the same factor.

Some properties of Pythagorean triangles include:
- Exactly one length value of a leg is divisible by 3.
- Exactly one length value of a leg is divisible by 4.
- Exactly one side-length value is divisible by 5.
- At most one length value of a leg is a square or a multiple of a square.

Here are the first few triples for m and n between 1 and 4
  n = 1 n = 2 n = 3 n = 4
m = 2
m = 3
m = 4
m = 5
m = 6
m = 7
m = 8
m = 9
m = 10
[3, 4, 5]
[8, 6, 10]
[15, 8, 17]
[24, 10, 26]
[35, 12, 37]
[48, 14, 50]
[63, 16, 65]
[80, 18, 82]
[99, 20, 101]

[5, 12, 13]
[12, 16, 20]
[21, 20, 29]
[32, 24, 40]
[45, 28, 53]
[60, 32, 68]
[77, 36, 85]
[96, 40, 104]
[7, 24, 25]
[16, 30, 34]
[27, 36, 45]
[40, 42, 58]
[55, 48, 73]
[72, 54, 90]
[91, 60, 109]
[9, 40, 41]
[20, 48, 52]
[33, 56, 65]
[48, 64, 80]
[65, 72, 97]
[84, 80, 116]

Pythagorean Triple Calculator

Enter two positive integer values:
m: > n:

A Pythagorean Triple:

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