Since we're doing dumb physics/maths questions...

[TimberWolf]

Put it on the back (stern) of the boat, pointing backwards (astern) so it pushes air out behind the boat. Additionally, take the sail down.

It's because of Newton's 3rd law, but I won't spoil the thread completely by explaining the reasoning just yet.

I'm wondering whether you could power the boat with the sail, not least because the fan will be carp. I'm not a boating person, but I'm imagining a triangular sort of shape of sail that's pivoted on one corner. If that's swung to and fro, it will push the boat backwards. Just keep flapping, and use the steery thing (tiller?) too.

[d000hg]

It's outside the scenario - this is a pure physics problem rather than a lateral thinking one - but yes in reality that would be quite effective, if rather tiring. You're translating biological (chemical) energy into kinetic energy to move the sail, effectively 'rowing' the boat with a 'paddle' in the air.

[Sockpuppet]

Feck all your high powered physics malarky.

If it were me and I wasn't allowed to point it at myself I would.

Stick it on the front and mount it backwards at a 45 degree angle pointing it towards the mast.

This will provide backwards thrust from the fan and also will allow the sail form as an aero foil and generate motion from that effect.

The sail will have drag but not 1:1 with the air pushed so it'll move the boat.

[TimberWolf]

It's outside the scenario - this is a pure physics problem rather than a lateral thinking one - but yes in reality that would be quite effective, if rather tiring. You're translating biological (chemical) energy into kinetic energy to move the sail, effectively 'rowing' the boat with a 'paddle' in the air.

All forms of propulsion will boil down to the same basic principle and that is throwing mass (or rather momentum) in the direction opposite you want to travel. You can do this using little mass and lots of energy (e.g. light) or using a more mass and less energy. Fans are more on the high energy side because they are quite small and push relatively little air out at high speed. Wings (and sails) are better because they push on more air. Rowing an oar in water for instance is going to be more energy efficient than rowing in air, unless you have a huge oar.

[xoggoth]

I'd go with the stick it in the water pointing backwards idea but how far are a few batteries going to take you anyway? Probably best to chuck them overboard and reduce the pointless weight.