Ed:
Great problem, thank you. After about five minutes I found a solution which
includes the 10, 17, 15, and 13 squares as corner placements.
Bob Wainwright
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clockwise, with corners in bold I get
17,16,13,11,15,14,7,10,12
Brian
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The corner squares are 17 13 10 and 15.
Some parts of my fill must be different than yours because I got different
groups of leftover squares.
Joseph DeVincentis