Polygon dissections

[JEK595]

A few dissection puzzles:

1. Dissect three identical squares like so: Divide one into three unequal parts, divide on into two unequal parts, and leave one whole. Then arrange those parts to form one larger, perfect square.

2. Dissect a Greek cross into nine pieces that can fit together to form either five small squares or one large one.

3. Divide a perfect pentagonal star into twelve pieces which can be rearranged to form four smaller, perfect pentagonal stars.

4. Divide a perfect pentagonal star into eighteen pieces which can be rearranged to form five smaller, perfect pentagonal stars.

5. Divide a perfect square into five pieces which can be rearranged into either a trapezoid with two right angles, a right triangle, a perfect Greek cross, or a parallelogram.

6. Divide a perfect nonagon into fifteen pieces which can be rearranged into three smaller, perfect nonagons.

------------------
-grace the mind with puzzles, games, and tapioca.-
-If at first you don't succeed, try, try again. (Not suitable for skydivers.)

[JEK595]

bump

C'mon. Try 'em.

[groza528]

what's the definition of a 'Greek Cross?'

[impossibleroot]

Believe me, I *am* trying the first one and it's killing me! I am learning a lot about dissections in general though, esp. that hinged dissection (triangle into square) -- very cool.

Don't tell me how, but is there a general way to solve these or do you just have to mess with it till you've got it?

[ralphmerridew]

A greek cross:

code:

---

   +--+


   |  |


+--+  +--+


|        |


+--+  +--+


   |  |


   +--+

---

Or, a square, plus a congruent square attached to each side.

[Jen Aside]

1. sounds like a tangram, but I'd have to remember what the composition is to tell for sure.

[groza528]

2. I'm pretty sure this works. First the easy part: dissect the cross into 5 identical squares (one at each 'arm' and one in the center.) Then dissect four of these squares in the following way: the cut goes from one corner to the midpoint of a non-adjacent side. The five identical squares are obvious, to get the large square I'll try to put it in code.

code:

---

 ----------------


|.       .   ....|


| .      ....    |


|  . ....  .     |


|....       .    |


|    .       ....|


|     .  .... .  |


|    ....      . |


|....   .       .|


 ----------------

---

Hope that's understandable

[Milt]

I tried number 1.



Any hope of a clue?

[Coyote]

Set the three squares side by side to make a 1x3 rectangle (ABCD)
Mark point E 1.7321... units from A (SQRT 3)
Mark point F SQRT 3 from D
Cut from C to E, making triangle ACE
Cut from F to G (which lies on line CE), making triangle CFG
Slide triangle ACE along line CE until C meets G
Move triangle CFG up so that CF lies on EB

code:

---

                              E 


A ----------------------------+--------------------- B


 |                |                |                |


 |                |    +           |                |


 |                |    G           |                |


 |                |                |                |


 |                |                |                |


 |                |                |                |


 |                |                |                |


 |                |                |                |


C ---------------------+----------------------------- D


                       F

---

Hope that made sense.

[Milt]

Thanks Coyote!

[This message has been edited by Milt (edited 03-04-2002 12:47 PM).]