Checkerboard

[dureisseix]

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.archives-ouvertes.fr/hal-01380815
It has been outperformed (2009) in
https://dspace.mit.edu/openaccess-disse ... 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 (half-folded 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??

[anonymous person]

This is really fascinating work - have you considered submitting a paper to 7OSME? http://osme.info/7osme/

[dureisseix]

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 9-year delay discussion (2000, 2009, now 2018 for 7OSME...)

[dureisseix]

Dear all,
Few years after, an update on the subject.
Seems that Jason Ku broke the 9-year 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?

[kareshi]

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.

[dureisseix]

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 up-to-date:
http://www.budaiorigami.hu/en/chequered

[kareshi]

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/