The Height of Women
(or: No Repeated Difference)

Five women are standing on platforms on a stage.  The shortest woman measures 5 feet, 0 inches.  Each woman is a different height, and the difference in height between any pair of women is a different integral number of inches.  There are four platforms on the stage, raising the shorter women to the height of the tallest.  If the total platform height is a minimum, what are the heights of the other four women?

Extensions: Some obvious extensions suggest themselves.  Repeat the problem for 3,4,6,7,8 women.  Is there any pattern or procedure which helps in the solutions?

Source: Original.

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