Date: Jul 12, 1995 9:54 PM
Author: Bob Hesse
Subject: Solution of Elephant Puzzle

Solution to Elephant Puzzle

Note: There will be an HTML version of this file sometime at the
Geometry Center's www server.  In the mean time, all images can
be found in the Geometry Forum's anonymous ftp site in the
/pictures/articles/ElePuz directory.

Here is one solution to the Elephant Puzzle.  I probably would not
have found this solution without the help of Lori Thomson.  First
mark the heads, tails, and bodies in the following manner.  The
image "Sol1" which shows this notation can be found at

ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol1.gif
or
ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol1.ps

Note that the head faces are labeled with numbers, the body faces
are labeled with lowercase letters, the tail faces are labeled with
uppercase letters, and the blank face is not labeled at all.

One solution is tail 'A', body 'e', head '1', and the blank face as
the bottom side.  To get this solution first fold the back of face
'A' onto the back of face 'b'. Next fold the front of face 'a' over
the front of face '2', the back of face '2' over the back of face
'B', and then the back of face '1' over the back of face 'a'.  After
doing these foldings, you should have the image "Sol2" found at

ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol2.gif
or
ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol2.ps

In this picture, the front of faces '1' and 'A' are visible when the
paper is flipped over.

Next fold the front of face 'b' onto the front of face 'c'.  After
doing this folding, the three elephant faces that we want are aligned
in a row as the figure "Sol3" found at

ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol3.gif
or
ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol3.ps

Lastly, fold the back of face '3' over the back of face 'c', the back
of face 'C' over the back of face 'd'.  Now the foldings are complete,
and the remaining four sides on top will fold into a tetrahedron where
the head, body and tail connect properly, and the remaining face is a
blank.  The final image, "Sol4" shows the desired four triangles and can
be found at

ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol4.gif
or
ftp://forum.swarthmore.edu/pictures/articles/ElePuz/Sol4.ps

In my posting of the puzzle, I had two additional questions:

        Is there more then one head, body, tail combination?
        How many solutions are there if the blank side must be a blank
        face?

Unfortunately, I have neither a satisfying answer to the first question,
nor a complete answer to the second one.

Yes there is another head, body, tail combination.  Again using the
notation above tail 'A', body 'd', head '1' with the backside of face '3'
is another solution.  But this answer is not satisfying to me since it
uses the same head and tail as in the described solution above.

If the blank side must be the blank face, then the body 'e' must also be
one of the faces in the solution.  So far the only solution I can find
that allows the inclusion of these two faces is the folding given above.

I think it is possible to answer these questions by listing possible
combinations and ruling the impossible ones out.  This is the approach
Lori Thomson was using to solve the puzzle and how I stumbled upon an
answer.  For instance if we list all of the head and tail combinations we
get nine possiblilities:

        1A 2A 3A
        1B 2B 3B
        1C 2C 3C

Both 1B and 2B can be immediately ruled as impossible, since the head
and tail combinations share an edge which forces the faces to be aligned
improperly.  Also 2A and 3A appear to be impossible combinations.  After
that I'm not sure if any other combinations can be ruled out.  So to try
and answer this puzzle, I would next list and consider the head and body
combinations, and then body and tail combinations.  Hopefully this
systematic listing and ruling out combinations we leave only a small set
to consider.

Does anyone have a more elegant way of finding solutions?  Although I
sometimes enjoy doing things that involve brute force, I would love to
see a slicker approach to the puzzle.