Department of Mathematics
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A \( (p, t) \) arrangement consists of \( p \) points and \( t \) (straight) lines in the Euclidean (or in the real projective) plane chosen so that each line has exactly \( 3 \) points on it. The classical orchard problem raised by Sylvester in [4] is to find an arrangement with the greatest \( t \) for each given value of \( p \). For example, Grünbaum et al (see [1]) have shown that \( t(12) = 19 \), \( t(16) = 37 \) and \(t(18) = 46 \). Concerning \( t(13) \), it is only known that \( 22 \le t(13) \le 24\), see {2], page 8. Since every line has 3 points in an orchard, the cubic curve set-up is ideal for studying problems in this area of discrete geometry. The points of the orchard lie on selected non-singular cubic curves. We use the group structure available on the cubic to show that resulting design has \( 19 \) collinearities which is exactly the greatest number of lines for this orchard as mentioned above. Aa an application, we exhibit the Orchard \( (12, 19) \) as a configuration in the projective plane over \( GF(11) \) where all the points and lines of the orchard are explicitly shown. Also, we give a one-parameter family of rational orchards \( (12, 19) \) i.e. a global realization with all the \( 12 \) points and the \( 19 \) lines having rational coordinates. Similar cubic curve representations are given for the orchard \( (18, 46) \).

1. Burr, Stefan A; Grünbaum Branko and Sloane, N.J., The Orchard Problem, Geom. Dedicata 2 (1974).I
2. Grünbaum, Branko., Configurations of Points and Lines, Graduate Studies in Mathematics vol 103, AMS, 2009.
3. Lanyon, Jeffrey and Padmanabhan, R., Group Realizations of some Orchards.
4. Sylvester J. J. Problem 3014, Mathematical Questions in Educational Times, 1894.