The Looking Glass Club gruff davies New Scientist

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How the checksums work

... and why you can't "brute force" the answers using the them.



Each solution is a 32x32 bit image, in other words 1024 bits.


There are therefore 2^1024 possible images.


The checksums perform an "exclusive OR" operation with each 32-bit row of the correct image, producing a 32-bit number, where each bit is essentially the parity of the bits in that column*.


For every checksum, there are 2^(1024 - 32) images that will match.  That's about 4 x 10^298.  This is an unbelievably large number†.  


Yet for every image, the chance of it having a given checksum is 1 in 4,294,967,296 so the chance of you accidentally coming up with a matching image is very slim.  It's therefore a nice way to check your answer.


If the checksum matches and your picture looks right (trust me, you'll know if it does) then you stand a good chance of having the correct solution.  


Now stop thinking about cheating and get solving the puzzles properly!

*Actually, each row is bitwise rotated by its row number prior to XORing to mix things up a bit and help prevent trivial false positives, so it's actually  the parity of the diagonals.

†For reference, there are only about 10 ^ 80 atoms in the universe.  

  Even Douglas Adams would have struggled to describe how unbelievably big it is.

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