Divide a side of the square into 8 equal segments.
Source: 1. rec.puzzles, 2-5. Original. Used as Ken's Puzzle of the Day
4/13/94.
Solutions were received from many sources.
Denis Borris pointed out that this puzzle is in the rec.puzzles archive
under
square.five problem.
That solution is similar to the one I have:
- Label square ABCD clockwise.
- Extend AD and BC to lines ADE and BCF.
- Draw AC and BD and label center O.
- Choose any point G on CD. Reflect G to H on AD: Draw AG, intersect BD at J, and extend CJ to H.
- Reflect G to I on BC.
- Choose another point K on CD. Reflect K to L on AD and M on BC.
- Extend HG to N on BF. Extend IG to P on AE.
- Extend LK to Q on BF. Extend MK to R on AE.
- Draw NP and QR, intersecting at X.
- OX bisects CD at S.
- AX and BX trisect CD.
- Reflecting S to the other sides creates the four squares.
- Drawing diagonal of the smaller squares leads to dividing one side into eighths.
Draw a trapezoid with one base 5/8 of the square side and the
other base one side of the square, find the intersection T of the
trapezoid's "diagonals", and reflect the five 1/8 sections through
T to 1/5 sections on the opposite side. Reflect the 1/5 sections
through O and draw five rectangles (solving part 1).
Thane Larson provided this
different construction. (I transcribe his drawing here):
- Draw AC and BD to find center of the square O.
- Extend AC beyond C and pick two points (E,F) on
AC outside the square.
- Draw BF and DE to cross at G. Draw BE and DF to cross at H.
- GH is parallel to BD (similar triangles). Extend GH to cross
(extended) BC at I and AD at J.
- BIJD is a parallelogram. BJ and ID cross at center K.
- OK bisects CD.
Thane and several others then disected the square into five equal
areas in this way. Bisect all four sides, and connect the midpoints
to non-adjacent corners, clockwise. This results in a central
square and four right triangles, all with the same area.
Other solutions were also received from Nick Baxter and Phillipe
Fondanaiche.
Denis Borris
also provided this slightly humorous solution:
Sticking to the wording of the problem (what is not explicitly
prohibited is allowed: per James F Fixx in his book "Solve It"),
my square is a one piece square.
Using it to trace squares of same size, I patiently make a 9 by 9 grid
on a large sheet of paper (at least 81 times the size of my square!).
I then use the middle square to be subdivided in 5 equal areas.
This is easily done by joining a corner of my square to the top
corner of the squares in first and last columns of the grid;
let me illustrate using part of the 9 by 9 grid, my square being ABCD:
L-----------------U
| | | | | | | | | |
M-----------------T
| | | | | | | | | |
N-----------------S
| | | | | | | | | |
O-------D-C-------R
| | | | | | | | | |
P-------A-B-------Q
Join AR AS AT AU, then BO BN BM BL;
this will "mark" CB and DA in beautiful "fifths".
So the job is completed.
Why do I use a 9 by 9 grid?
Well, the "joining" can be similarly continued,
marking DC and AB in similar "fifths", hence
resulting in my square being subdivided in
twenty-five (25) equal areas!
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