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Most books have an ISBN number.  For example, the ISBN number of Recreations in the Theory of Numbers is 0486210960.  Let's multiply these digits by the numbers ten to one.  10*0+9*4+8*8+7*6+6*2+5*1+4*0+3*9+2*6+1*0 = 198 = 18*11.  The sum is divisible by 11.  Two of the Martin Gardner books are 0879752823 and 0226282546.  Using this same method, we get the sums 286 (26*11) and 165 (15*11).  So far, all of the sums have been divisible by 11.  But how about the ISBN number of The Book of Numbers -- 038797993X ?  We'll let the X represent 10, and we obtain the sum 308 (28*11).  For any book you pick, this method will yield a number divisible by 11.  Makes a great bar bet in bars with books.  The ISBN's are designed that way. The final digit (or X) is a checksum.  The number 0100000001 cannot be an ISBN number.  If we know that only one digit is wrong, then the correct ISBN number might be 0600000001, 0170000001, 0108000001, 0100200001, 0100090001,  0100003001, 0100000401, 0100000061, or 0100000002.

William Ivey, Greg Bensimon, Derrick Schneider and Rod Bogart pointed out that my previous mentions of ISDN should be ISBN.  Claude Chaunier and Joe DeVinctentis solved my puzzle about ISBN's.  123456789X is the only valid starting ISBN for the puzzle.  Since it's a 'good' ISBN, the problem as stated has no solutions.

Data transmission can't afford to go back and check too often.  With the ISBN, a wrong digit will let you know that a mistake was made, but you won't know what the original data was.  Most data transmissions use what are known as error-correcting codes to ensure data integrity over noisy lines.  A cellular phone transmission or your viewing of this page over the internet both involve an error correcting code somewhere along the line.  If your data-receiving device detects an error due to noise, it can correct the data.

In 1949, M. J. E. Golay published an article called Notes on digital coding in the Proceedings of the Institute of Electrical and Electronic Engineers.  It was a very short article, a mere half a page.  Despite it's length, the article turned out to be one of the most important publications of this century.  Today, reliable data transmission uses methods developed from that original article.  Notes on digital coding described what is now known as the Golay code.

 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0
One of many generator matrices for the extended [24, 12, 8] Golay code
Suppose I wanted to send the message 010001010010.  I would do a even/odd parity check of the second, sixth, eighth, and eleventh rows to obtain 010001010010 010000001110, and send it.  As long as noise caused fewer than three errors to my data string, the receiver would be able to reconstruct my original data.  For example, if 110001110010 011000001110 was received, the correct message would be quickly discerned.

It has a nice application, then.  What else?  The game of Turning Turtles can be played with 24 coins, heads up.  Each move, a player must turn one coin tails up, and may also turn over up to six other coins to the left of the first one.  The other six or fewer coins, unlike the first, may be turned either heads up or tails up.  The winning strategy is to always return to a valid Golay code.

The Golay code is a perfect code.  The only other nontrivial perfect codes are the Hamming codes.  A perfect code is such that it's packing radius is equal to it's covering radius.  In this case, the radius is 3.

Packing above relates to data, but another property of the Golay code involves sphere-packing in the 24th dimension.  Spheres can be packed perfectly in the 24th dimension via the Leech Lattice, which is directly constructable from the Golay code.  In group theory, the important Mathieu Groups are closely related to the Golay code.  The Golay code contains an unmatchable amount of symmetry.  Pretty amazing for a half-page paper.