Cutting a hexagon into 6 congruent pieces
Vikram Sidhu
on the assignment page):
It's my birthday and I pride myself in being original, but I need your
help figuring out how to divide pu my birthday cake. The cake is a
HEXAGON and the pieces need to be congruent, so that nobody complains.
There will be six people eating the cake. HOWEVER, the pieces cannot
be equilateral triangles, since that is not very interesting.
Give me at least 2 ways I could divide the cake. Use pictures, words
and symbols to clearly explain each method.
Thanks for taking the time to read this problem and hopefully, you
will be able to help me out.
THANKS!
Mary Krimmel
Think of a jigsaw puzzle with interlocking pieces, or of Escher-like tile
designs. Be as imaginative and interesting as you please. Or if you imagine
actually having to make the cuts, be as plain and straight as you please.
Or for an entirely different concept, think of the hexagon as being
composed of congruent diamonds. Then you have not only the "not very
interesting" and hence forbidden division of each diamond, but what other
possible division, hardly more interesting but somewhat less obvious.
(In fact, I personally find an equilateral triangle both beautiful and
interesting.)
Enjoy, and best wishes.
Mary Krimmel
ma...@krimmel.net
Art Mabbott
Tx several years ago. If you find the center of the hexagon and then cut a
line to any point along the first side and then do a congruent cut to the
same corresponding point along each of the other 5 sides, all of the areas
(surface area of the tops) will be congruent and all of the hts will be
equal and so all of the volumes (the pieces) will be the same. The first
cut that you make doesn't have to be a straight one...just as long as each
of the cuts are "congruent" Proving that they each get the "same" piece is
fun (same means the same volume of cake and the amount of frosting (on top
and on the sides)) - especially when you have more people than you have
edges.
Art Mabbott
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Dan Hoey
>(as written on the assignment page):
No, thank you. Assigners of problems should either provide help
or forbid it. And if they do neither, they should at least direct
requests for assistance not to include "puzzles" groups, which
suffer from the contrasting aims of our intellectual interest, your
education, and other assignees' recognition of their accomplishment.
That's my view, and it may or may not be popular, but I decline to
debate it; I'm just not that interested. I want to address the
mathematical point implicit in the misstatement of your problem.
> It's my birthday and I pride myself in being original, but I
> need your help figuring out how to divide [up] my birthday cake.
> The cake is a HEXAGON and the pieces need to be congruent,
> so that nobody complains.
The setter (or transcriber) seems to have forgotten that not all
hexagons are regular. I expect that many hexagons cannot be
divided into six congruent pieces, but I don't have an example--
anyone care to give one (with proof)?
An extended question is to characterize the hexagons that can be so
divided. We might even extend this to six-segment closed paths that
may be permitted to intersect themselves, so that "divisions" like
I've drawn below might occur (where I hope you can figure out what
happens at the (#) marks, which are too hard to draw in ASCII).
_______
\ : /
\ : /
\:/
X
/:\
# : #
/ `*' \
______/#######\______
\ .-/ \-. /
\' / \ `/
\/ \/
Enjoy!
Dan Hoey
hao...@aol.com