# Maximal zero sums in an order 4 projective plane, using -10 to 10 for the points

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.

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"21 vectors each with 5 variables" I don't see any vectors, nor any variables. What do you mean? –  Gerry Myerson Jul 5 '11 at 5:41
The statement of the question is very unclear. Here's what I've gathered from trying to fathom it; if this is correct, I'd suggest you amend the question accordingly to make it less confusing: The number are redundant; the number squares are the same as the letter squares (if the letters are counted from 'a' $\leftrightarrow$ 1). The letters stand for the points of a projective plane of order $4$. The lines of the plane are defined by the rows, columns and main diagonals of the two squares (where each of the two occurrences of a letter stands for the same point)... –  joriki Jul 5 '11 at 7:29
... One of the points corresponding to the letters 'q' to 'u' is added to each of these lines as indicated, but the rows and columns formed by these additional letters don't correspond to lines of the plane; instead, an additional line containing all of the five additional points labeled 'q' to 'u' is added to complete the plane. Then, when you look for "the maximal number of zero sums", you mean that you consider all possible assignments of the numbers $-10$ to $10$ to the points of the plane and then for each line you form the sum of the numbers assigned to the points on the line... –  joriki Jul 5 '11 at 7:31
... and you want to know the maximal number of lines with sum $0$ under any such assignment. Correct? –  joriki Jul 5 '11 at 7:34
Thanks, joriki, that sounds like what Ed is trying to do, and I hope he'll edit accordingly. When you write "main diagonals," you want both upper-left to lower-right and upper-right to lower-left, don't you? That's the only way I can see 21 lines here. –  Gerry Myerson Jul 5 '11 at 11:01