The solution given in the OP does explain where it came from: it says the 2nd relation applies to consecutive Fibonacci numbers (this follows from the Cassini identity). The difference of these is the Fibonacci number that comes before them, and it is required to be a square. The only square Fibonacci numbers are 0, 1, and 144. Only 144 is 3 digits, and b and a must be the next two Fibonacci numbers: 233 and 377.
Chronos' approach is headed in the direction of Fibonacci numbers too, since ratio of the two values is very close to the golden ratio.
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