There have been a short discussion about the checkerboard (the board, not the chess pieces):
http://snkhan.co.uk/forum/viewtopic.php ... ard#p28670
but this post concerns a more general challenge: folding the nxn checkerboard from a single square sheet of paper with one color per face. Additionnaly, the question is: what is the optimal design (i.e. the smallest initial square for the same folded design)?
An old (2000) upper bound on the paper size has been stated in
https://hal.archivesouvertes.fr/hal01380815
It has been outperformed (2009) in
https://dspace.mit.edu/openaccessdisse ... 21.1/62156
but only for n > 16. Since then, there have been several effectively folded models of the expected optimal chessboard (n=8), but I have not seen real models folded for a larger n.
So, here is my latest attempt: the 11x11 checkerboard
from a 60x60 paper (halffolded to feel the complexity).
The design principle is not new, and is described in
https://arxiv.org/abs/1510.07499
but the real physical folding may well be.
So, the question is:
the effective folding of the optimal 11x11 checkerboard (i.e. from the smallest initial square of paper  not proved but strongly suspected), is it a world record??
Checkerboard
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 Senior Member
 Posts: 437
 Joined: April 14th, 2010, 11:54 am
 Location: London
Re: Checkerboard
This is really fascinating work  have you considered submitting a paper to 7OSME? http://osme.info/7osme/
http://www.flickr.com/photos/arunori/
Simplifying is complex
Simplifying is complex

 Newbie
 Posts: 6
 Joined: September 11th, 2017, 4:52 pm
Re: Checkerboard
For the submission to 7OSME, the answer is definitely: yes. The working document on arxiv was made for steering me to work on it. Moreover, this is 9year delay discussion (2000, 2009, now 2018 for 7OSME...)

 Newbie
 Posts: 6
 Joined: September 11th, 2017, 4:52 pm
Re: Checkerboard
Dear all,
Few years after, an update on the subject.
Seems that Jason Ku broke the 9year delay between new results...
Now he says that the strongly suspected optimal bound for n less than 16 was false!
Indeed, he succeeded in folding a 8x8 chessboard (so n=8) from much less than a 32x32 square as previously done: he broke the bound to 26x26 square!
Our challenge for the moment is to fold the model from the CP he provided:
original one at http://jasonku.mit.edu/26board1.html
simplified version at https://twitter.com/origamimagiro/statu ... 3524363264
These date back to 2020... Does someone have newer information than I have?
Is there a diagram for this model?
Is there an even better new bound?
Few years after, an update on the subject.
Seems that Jason Ku broke the 9year delay between new results...
Now he says that the strongly suspected optimal bound for n less than 16 was false!
Indeed, he succeeded in folding a 8x8 chessboard (so n=8) from much less than a 32x32 square as previously done: he broke the bound to 26x26 square!
Our challenge for the moment is to fold the model from the CP he provided:
original one at http://jasonku.mit.edu/26board1.html
simplified version at https://twitter.com/origamimagiro/statu ... 3524363264
These date back to 2020... Does someone have newer information than I have?
Is there a diagram for this model?
Is there an even better new bound?
Re: Checkerboard
Hello! I’ve been challenging myself with checkerboards this year. The results are on my Instagram @danbrownorigami
So far I have solved for the following, all are seamless (No folded edge goes through the individual squares) 
4x4 from 9x9
5x5 from 13x13
6x6 from 18x18
7x7 from 25x25
8x8 from 30x30
9x9 from 43x43
11x11 from 65x65 *I believe I have it solved for 61x61 but haven’t made it yet.
I don’t have a seamless n solution, however each of these have similar structures. I haven’t solved for 10x10 yet. Odd ones let me make the same corners all around, so I think I’ll even try 13x13 before 10x10 or 12x12.
As for further updates, I believe a new online catalog of checkerboards is in process.
So far I have solved for the following, all are seamless (No folded edge goes through the individual squares) 
4x4 from 9x9
5x5 from 13x13
6x6 from 18x18
7x7 from 25x25
8x8 from 30x30
9x9 from 43x43
11x11 from 65x65 *I believe I have it solved for 61x61 but haven’t made it yet.
I don’t have a seamless n solution, however each of these have similar structures. I haven’t solved for 10x10 yet. Odd ones let me make the same corners all around, so I think I’ll even try 13x13 before 10x10 or 12x12.
As for further updates, I believe a new online catalog of checkerboards is in process.

 Newbie
 Posts: 6
 Joined: September 11th, 2017, 4:52 pm
Re: Checkerboard
Wow. Impressive. Indeed the seamless constraint is more difficult to tackle.
We will need to send a message to Peter Budai to make his web page uptodate:
http://www.budaiorigami.hu/en/chequered
We will need to send a message to Peter Budai to make his web page uptodate:
http://www.budaiorigami.hu/en/chequered
Re: Checkerboard
Small update from the above that my seamless 8x8 is down to a 28 grid and the 9x9 is now a 39 grid.
Update (10/1/24) the seamless 11x11 is now 57 grid.
Jason Ku’s new list is live here:
https://origamimagiro.github.io/checkerboards/
Update (10/1/24) the seamless 11x11 is now 57 grid.
Jason Ku’s new list is live here:
https://origamimagiro.github.io/checkerboards/