Also, here is a hint on puzzle 1, you could read first:
I believe the above answer is optimal. It only passes through natural numbers in its intermediate computations. I believe it is impossible to match the score of the above solution if you limit yourself to natural numbers below a googol.
This comment is the best I can do on puzzle 1. Read it only if you want to be spoiled.
Also, here is a hint on puzzle 1, you could read first:
I believe the above answer is optimal. It only passes through natural numbers in its intermediate computations. I believe it is impossible to match the score of the above solution if you limit yourself to natural numbers below a googol.