Did he stop in front of the mountain?

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No! He stopped in the town, not in front of the mountain.

Did he finish his measurement when he stopped there?

Yes! He had a mountain of information at his fingertips.

Did he know the exact distance between the mountain and the place he stopped at?

There is no information given in the puzzle about the man knowing the exact distance between the mountain and the place he stopped at.

Did he use his fingers for the measurement?

No! The man did not use his fingers for the measurement.

He counted something with his fingers?

No! The man did not use his fingers for the measurement.