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q  abcd   idfo  t     q  efgh   gnla  t   1  2  3  4    9  4  6 15   The two 4x4 squares are part of  q  ijkl   pecj  t   5  6  7  8    7 14 12  1   a resolvable block design.  Any q  mnop   bkmh  t   9 10 11 12   16  5  3 10   two numbers are on one column,                     13 14 15 16    2 11 13  8   row, or main diagonal. s  rrrr s uuuu  s   

The 2 4x4 squares give 16 variables with 20 lines of 4 in the rows, columns, diagonals. Adding 5 more variables gives an order-4 projective plane, with qrstu as the 21st line. Just to be clear, the 21 lines in the first representation are qabcd qefgh qijkl qmnop aeimr bfjnr cgkor dhlpr idfot gnlat pecjt bkmht igpbu dneku flcmu oajhu dgjms afkps olebs inchs qrstu. Another order-4 projective plane uses the lines mod((1,2,5,15,17)+n,21) for n=0-20.

If the 21 variables are labeled -10 to 10, what is the maximal number of 0 sums that are possible? I've gotten many arrangements with 11 zeros, but I've found no arrangements with 12 zeros.

order 4 prejective plane

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    I've made some edits. And yes, I'm wondering about the maximal number of zeros when the 21 points are labeled with$-10$to 10.2011-07-06

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