Great job mates (particularly Ruud)!

I got 18, so 2 more for you guys & gals to discover...

But there could be more that I haven't realised...

Anyway you could try to list out the "area ratio" of each configuration, e.g.: 1,1,1,1 or 1,3,12,48...

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The truth is, you probably could not find any significant inspiration by merely looking at the previous 17 configurations... But #17 (from Ruud) could give some slight glimpse of hope, especially if you can guess how Ruud came up with that...

In fact I thought Ruud could have nailed it when he found #17 since #17 & #18 are normally found together...

Anyway I'll wait one more day and preferably with Ruud's formal approval to show the final one out... Or if somebody can work it out from my hints above...

In fact I thought Ruud could have nailed it when he found #17 since #17 & #18 are normally found together...

Anyway I'll wait one more day and preferably with Ruud's formal approval to show the final one out... Or if somebody can work it out from my hints above...

- udosuk
**Posts:**2698**Joined:**17 July 2005

Some graphics done by me...

Bi-cutting 3,4,9

#01: 9,12,16,27

#02: 3,9,16,36

#03: 1,3,3,9 (a)

Bi-cutting 1,3,12

#03: 1,3,3,9 (a) (duplicated)

#04: 3,4,9,48

#05: 1,3,12,48

Bi-cutting 1,1,1

#06: 1,3,4,4 (a)

#07: 1,3,4,4 (b)

#08: 1,3,4,4 (c)

Tri-cutting 1,3

#09: 1,1,1,1 (a)

#10: 1,1,1,9

Flipping rectangle of 1,3,3,9 (a)

#11: 1,3,3,9 (b)

Rotating equilateral triangle of 1,1,1,1 (a)

#12: 1,1,1,1 (b)

#13: 1,1,1,1 (c)

Flipping rectangle of 1,1,1,1 (b)

#14: 1,1,1,1 (d)

Tri-cutting 30-30-120 triangle of 1,2

#15: 1,1,3,4 (a)

#16: 1,1,3,4 (b)

triangle within triangles

#17: 1,4,4,16

#18: 9,12,12,16

I deliberately tried to make good approximation to the 30-60-90 triangles...

I regarded 15-26-30 as the best "unit" side ratio...

Bi-cutting 3,4,9

#01: 9,12,16,27

#02: 3,9,16,36

#03: 1,3,3,9 (a)

Bi-cutting 1,3,12

#03: 1,3,3,9 (a) (duplicated)

#04: 3,4,9,48

#05: 1,3,12,48

Bi-cutting 1,1,1

#06: 1,3,4,4 (a)

#07: 1,3,4,4 (b)

#08: 1,3,4,4 (c)

Tri-cutting 1,3

#09: 1,1,1,1 (a)

#10: 1,1,1,9

Flipping rectangle of 1,3,3,9 (a)

#11: 1,3,3,9 (b)

Rotating equilateral triangle of 1,1,1,1 (a)

#12: 1,1,1,1 (b)

#13: 1,1,1,1 (c)

Flipping rectangle of 1,1,1,1 (b)

#14: 1,1,1,1 (d)

Tri-cutting 30-30-120 triangle of 1,2

#15: 1,1,3,4 (a)

#16: 1,1,3,4 (b)

triangle within triangles

#17: 1,4,4,16

#18: 9,12,12,16

I deliberately tried to make good approximation to the 30-60-90 triangles...

I regarded 15-26-30 as the best "unit" side ratio...

- udosuk
**Posts:**2698**Joined:**17 July 2005

If that's it for 4 (and it certainly looks and "feels" right) then what about 5? Once that is answered it ought to possible to conjecture a series or proof for 6, 7, 8 etc.

- underquark
**Posts:**299**Joined:**06 September 2005

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