There are an infinite number of great circles, but several reasonable simplificatons are to limit the problem to great circles which go through the poles (i.e., lines of longitude and their antipodal lines of longitude), and to discrete longitudinal steps.
The Great Circle Mapper's "Plain" maps have just three colors: water, land, and shore. Using the raw 3600 × 1800-pixel map (0.1° increments) it was a straightforward problem to count the number of pixels of each color in each one-pixel-wide vertical column of the map, then sum each column from the left half with its counterpart 1800 pixels to the right, then scan for the columns with the most land and most water.
The polar great circle including 70.0°W longitude and 110.0°E longitude covers the most land, 53.9% of its path:
The polar great circle including 17.5°W longitude and 162.5°E longitude covers the most water, 85.1% of its path:
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Karl L. Swartz.
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