"gripping and hugely entertaining... difficult to put down."
- New Scientist
Even read by Werner Voguls (Amazon CTO)
... 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.
© Sencillo Press and Gruff Davies 2010