From: Evan Whitney <evan@scd.hp.com>
Newsgroups: rec.puzzles
Subject: <SPOILERS>Isosceles Triangles Puzzle Revisited
Date: Wed, 19 Aug 1998 10:10:59 -0700

Yum, yum!! That's mighty fine crow! Please pass the BBQ sauce... :-)

Congratulations to Mauro Fiorentini of Italy who was the only person
to find a solution to the Isosceles Triangles Puzzle:

   What is the fewest number of angularly unique isosceles triangles 
   that can be pieced together to form a solid square?

   "Angularly unique" means that no two triangles can have the same
   set of three angles. An isosceles triangle contains at least two
   identical angles.

I found a solution to this puzzle using fewer than 6 triangles,
and challenged the world to find a different solution. Mauro was
the only one to send me a solution, and what do you know... it was
different than mine!  In lieu of $50 prize, Mauro has opted for
merchandise. "Jay... what do we have for our contestant this evening?"

   "Well, Evan, it's a collection of mathematical books by Martin
Gardner!" {Applause]

Both solutions work only for a specific set of angles, so it is
just possible that there are EXACTLY TWO solutions to this puzzle,
but I'm afraid our prize warehouse is empty, so if anyone finds yet
a third solution, you'll have to settle for fame and glory, only.

Both solutions share one common triangle. Here is a hint ROT13'd:

     bar natyr vf avargl qrterrf

Also, Mauro's solution contains two triangles that are angularly-
unique, but have the same area; whereas, each triangle in my
solution has a different area. Since "unique area" was not a
requirement Mauro's solution is perfectly valid. Is there more
than one solution that is angular- AND area-unique? Who knows??

Several folks were more interested in my code than in the puzzle.
I merely concatenated all of the digits of the unique angles of
each triangle and then added PI digit-for-digit to produce the
final code. So to decode:

 471672776613569
-314159265358979
================
 DDDDDDDDDDDDDDD  <- this is spelled out in more detail, below.


Both solutions follow 30 lines of <SPOILER> space. They are crude
ASCII drawings that must be viewed using a non-proportional font.
1
2
3
4
5
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9
10
11
12
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18
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Here is one way how to cut a square into 5 angularly-unique 
isosceles triangles:

+--------------------------------+
| \                            A |
|B  \                            |
|     \                          |  A : 90     45     45
|       \                        |  B : 45     67.5   67.5
|         \                      |  C : 112.5  33.75  33.75
|           \                    |  D : 135    22.5   22.5
|             \                  |  E : 157.5  11.25  11.25
|               \                |
|                 \              | Each triangle is labeled with
|                   \            | its letter placed closest to the 
|                     \          | triangle's apex (unique angle).
|                 ___/C \        |
|             ___/        \      | Each triangle's area is unique.
|         ___/.....         \    |
|     ___/D\___     .....     \  |
|____/         \____      ......\|
+-------------------\E-----------+

Here is "the other" way:

+-------------------------------+
|\                            A |
|B \                            |
|    \                          |  A : 90     45     45
|      \                        |  B : 45     67.5   67.5
|        \                      |  C : 112.5  33.75  33.75
|          \                    |  D : 22.5   78.75  78.75
|            \                  |  E : 157.5  11.25  11.25
|              \                |
|                \              | Each triangle is labeled with
|            _____/\            | its letter placed closest to the 
|      _____/      C \          | triangle's apex (unique angle).
|_____/                \        |
|     \_____             \      | D and E have the same area.
|          D\______        \    |
|      _____/   E  \_____    \  |
|_____/                  \_____\|
+-------------------------------+

	
The code is simply the digits of the angles added to
the digits of PI. To decode, subtract PI digit-for-digit
from the code as shown:

  471672776613569      code
- 314159265358979    - digits of PI
=================
  157513511254590 -> 157.5, 135, 112.5, 45, 90
                        E    D     C    B   A   (For first solution)

Evan Whitney
evan@scd.hp.com