Checkerboard Binary Calculations

Hi, I came across a Binary grid that has a checkerboard pattern. No two bits are ever next to each other vertically or horizontally, but they can be diagonally. The bits are staggered and they repeat over and over again. Something like below, where there is never a bit on one of the X blocks. The columns seem to have four bits down.
-----------
1X0X1X0
X0X0X0X
0X0X0X0
X0X0X0X
0X0X1X0
X1X0X0X
0X1X0X1
-----------

The problem is that it don't conform to the typical binary system of 1,2,4,8,16,32 etc.

I'm not sure what this is used for, but I was wondering if anyone had ever seen binary in a checkerboard pattern?

Thanks,
Fulano
Asked:
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Commented:
What are the X's?  I don't understand the description.
Author Commented:
The Xs are the areas of the grid that you cannot put a bit into. I've never seen a binary grid that has a "dead spot" for lack of a better word. You can change the bits in the 1 and 0 spots, but never in the X areas. That's what is puzzling to me. There is never any data in the X blocks.

Thanks,
Fulano
Commented:
Are the X's always 0 then, or just fixed data? Where is this data structure coming from?  Why is it 7x7?  If you read it across without the line breaks the X bits simply alternate.
Author Commented:
There is never any data in the X spots. They are dead spots. The only valid data areas are where you see either a 1 or a 0. That's why this is so puzzling to me. I've never seen this type of binary setup. i think its some sort of encrypted binary scheme of some sort, but I've never come across one like this.

Regarding the 7x7, that's just the number of rows and columns I used for a visual representation. That was purely arbitrary on my part. It can be much larger 7x15, etc.

I believe it contains a hidden number. I'm not looking to solve it, I'm just hoping someone can point me in the right direction of what scheme this might be.

My main question is: has anyone ever seen an alternating checkerboard pattern in binary, and if so, who do you calculate a value from that?

Thanks,
Fulano

Author Commented:
Typo -- who do you calculate a value from that? Shoud read

>how< do you calculate a value from that?

Commented:
I still don't understand the context of this: how do you address bit to get "no data".  But I had three hypotheses what it could be related to, one of them was hidden data as you suggested.  My other guesses are: packing bit fields into bytes where not all values are allocated (e.g. assigning space for four-digit values but only using two of the four possible values; monochrome GIFs do exactly this before compressing); or, some kind of sub-pixel rendering where the alternating unused bits would cause the result to be a different color (the seven-bit-across pattern reminded me of Apple ][ hi-res graphics which worked like this).
Author Commented:
I agree with you...I don't understand it either. It's a hidden code scheme used in Laser printers. That's where I saw this, and I found it to be really interesting. So, just I wanted to see what they were using as an encryption method, but I don't understand the checkerboard pattern. -- If you lay the grid out on paper, you see that the bits appear in a checkerboard pattern where no two bits are adjacent to each other vertically or horizontally, but they can be diagonally. So you get   X_X_X_X_X_X_X and the next row is _X_X_X_X_X_X_, etc, where the bits are in place of the underscores in my representation here. They also alternate.

What the heck type of binary code is that?

BTW, I don't believe this guy's interpretation of the bits is accurate. I think he's just guessing at the layout.

http://frotz.zork.net/pipermail/printers/2005-October/000048.html
Commented:
Are your data coming from the printer driver or are you reading back information from the printer itself?  This is interesting; I've read that printers have some kind of watermarking done in yellow so you can tell what printer it came from; maybe if that's true it has something to do with that.  The less suspicious interpretation could be that it has something to do with the printer trying to pretend it has better resolution than it actually has memory for.

I just Googled the watermarking issue and found a couple reliable sources:
http://www.eff.org/issues/printers
http://www.freedom-to-tinker.com/blog/felten/eff-researchers-decode-hidden-codes-printer-output
Author Commented:
You get it from a "physical" hard copy from the printer - an actual printed page and you read it off of that page. Each page is exactly the same. This is not an urban legend - it's real. The Xerox code has been decoded by the guys at EFF.org and that one was a very simple binary addition grid that you can see at the link below: (They explain how they decoded it in the article.)

http://w2.eff.org/Privacy/printers/docucolor/

However, the matrix used other printers is far more complex. The difference being that the Xerox grid allows for bits to be positioned in columns adjacent to each other vertically and horizontally, without a checkerboard pattern. Thus, allowing for simple binary representation (i.e. 1,2,4,8,16, etc). The other grid has a checkerboard layout - so how do you do the binary math? The same binary math does not work.

Which brings us back to my original question...have you ever, in all your binary experience, seen a binary layout that has a checkerboard pattern.

I too find it a huge mind buster puzzle. I enjoy math puzzles and this has been the most challenging I've every tried.

Thanks for your help,
Fulano
Author Commented:
Oh, one more thing...someone told me this may be called "Plural Bit Binary", but I don't really understand what that is. I know binary to be single but not "plural." Have you ever heard of that?
Commented:
Never heard of that.  All Google turns up is lots of patent applications, where it seems that "plural bit binary" is patent-lawyerese for a binary number longer than 1 bit.
Author Commented:
Hmm, interesting! OK, so I guess I'm still in search for a binary method that works with some sort of checkerboard pattern. Wow, they did a good job of encrypting this stuff so no one could read it. -- Not sure why, since all you can get from these codes are dates, times and serial numbers.

I'm also shocked of how little information is really out there about this. A lot of articles about the same thing, but no real hard information.

Thanks and if you think of something please let me know.

Fulano
Commented:
If it truly is encryption, I can imagine some possible reasons.  First, they might hope that if you can't figure out any encoded information, you'll assume it's random errors/ink-splots/compression rellics, or whatever, rather than leaking information.  Second, they might want to have some control over the information being discovered, even some amount of control over the government, so that some perceived privacy violation doesn't occur that will get people's attention.  I've read this a long time ago on web sites about printers encoding information in images, but I've never seen it in the mainstream media, and that is probably what the printer manufacturers want.

For any kind of binary arithmetic, I'd think you could either remove the blank bits entirely, or replace them with zeros, either shifting or not shifting alternate lines to make them line up.  If none of those prove useful, the encryption could be more complicated than just the arrangement of the pixels.

I'm curious how printers know what time it is.  Unless it is the printer driver sending these patterns, but that would be easier to defeat; you could disassemble the driver to figure out the encoding, or just use your own driver.

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Author Commented:
I'm not sure how the printers get their information about time and date. That would be interesting to know. I've never set the date and time on any of my printers, nor have I seen a manual that talks about that (there might be one, but I've never seen it). As for the S\N, that's probably straight out of the firmware.

Interestingly enough, Xerox didn't encrypt their pattern. Its just straight out binary addition. Something must have happened along the way that caused the other manufacturers to go to such a over the top approach.

I've also read where they've come up with signatures for images as well as ink jet printers, so I think this is a big business that somehow is under the media radar. Personally, I don't care...they can know my printer information all they want, but I'm sure there are some that would care if they knew.

If you come up with anything else, please let me know. - I'll keep the post open for a few more days to see if someone else comes up with anything that might help.

Thanks,
Fulano
Author Commented:
Superdave, I think I figured it out. Thanks for the help.

Fulano
Author Commented:
Good discussion. Thanks for the help!
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