While digging through the cavernous catacombs of my image directories, I found something interesting that I only vaguely remember stealing from some random website. I just hope I can do the problem by justice with an ASCII transcription.

Anyway, here goes. You take a simple right triangle, divided into sections like so:

1 |\
2 |a\
3 |aa\
4 |aaa\
5 |aaaa\
6 |aaaaa\
7 |aaaaaa\
8 |_______\
9 |bb|cccc|\
0 |bb|_ccc|d\
1 |bbbb|cc|dd\
2 |bbbb|cc|ddd\
3 |____|__|____\


Then switch those same sections around within the triangle:

1 |\
2 |d\
3 |dd\
4 |ddd\
5 |____\
6 |cccc|\
7 |_ccc|a\
8  _|cc|aa\
9 |b|cc|aaa\
0 |b|__|aaaa\
1 |bbbb|aaaaa\
2 |bbbb|aaaaaa\
3 |____|_______\


It doesn't transfer very faithfully into ASCII, but you can generate the same problem on a sheet of graphing paper with the left edge of the triangle being thirteen squares high and the bottom of the triangle being five squares wide. Section "a" is eight units high and three units wide, section "d" is five high and two wide. Sections "b" and "c" interlock on the first triangle, section "c" being almost identical to "b" except for having one less square in the wide section of the piece. However, when switching the whole thing around and keeping the shapes completely intact (which is sorta impossible for me to do with PRE tags unless there's some new fangled slash that has about a 15 degree slant that I don't know about), you lose an area of a square.

All right. Anyone care to explain why?