The path is not going to be an infinite spiral but that's sort of on the right track. The problem with that is that as Mario meanders close to the shore Kong will be able to just stroll around on the shore and Mario will never be able to safely disembark.

I believe the correct path will be an arc such that at every moment in time Mario is travelling in a direction that is directly opposite from Kong's position (instantaneously -- feels like calculus to me!). So, if Kong starts out deciding to run in a clockwise direction from 12 o clock to 1 o clock, then by the time he reaches 1 o clock Mario has curved his downward path, starting out by rowing towards 6 o clock but gradually curving his path so that he is now pointed in the same direction as a vector from the center point to 7 o clock would point. This makes it so that at every moment in time, Kong is indifferent about whether or not to continue running clockwise or counterclockwise, and if he switches then Mario just switches the direction of his arc. By the time Mario reaches the coast, he must be travelling fast enough so that this strategy still yields an arc that is outward as compared to an orbital path. (My hunch is that Mario could use this strategy but travel so slowly that Kong could simply run around in circles and Mario would never make outward progress beyond a certain radius).

As for calculating the actual number . . . that just hurts my brain!

Instant Edit: I think the above explanation is incorrect but I left it there to show the idea of what we are trying to achieve (Kong's indifference). We don't travel in the opposite "direction" from Kong (the opposite of his vector from the circle's center point to his position). Instead, we constantly travel in a direction towards the POSITION that is the opposite of Kong's POSITION. So, if Kong has travelled to the 1 o clock position, Mario has now curved his arc so that he is now pointed towards 7 o clock! Note that this direction of travel is NOT the opposite of the direction of the vector from the circle's center to Kong's position since Mario is no longer at the circle's center, nor is he positioned anywhere along the line from the circle's center to 7 o clock (he is on a curve towards that position -- the shape of the curve is NOT an arc of a circle . . . it's curvature is instantaneously becoming more -- "curvey?" as Mario travels farther and farther from the circle's center.

Calculating the number still hurts my brain!