Jaydeep Chipalkatti Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

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This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to six conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

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    Avritzer, D., and Lange, H. Curves of genus 2 and Desargues configurations. Adv. Geom. 2 (2002), 259280.

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    Baker, H. F.Principles of Geometry, vol. II. Cambridge University Press, 1923.

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    Chipalkatti, J. On the coincidences of Pascal lines. Forum. Geom. 16 (2016), 121.

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    Conway, J., and Ryba, A. The Pascal mysticum demystified. Math. Intelligencer 34,3(2012), 48.

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    Cox, D., Little, J., and O’Shea, D. Ideals, Varieties and Algorithms, 3rd ed. Undergraduate Texts in Mathematics. Springer, 2007.

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    Howard, B., Millson, J., Snowden, A., and Vakil, R. Adescription of the outer automorphism of S6, and the invariants of six points in projective space. J. Combin. Theory Ser. A 115,7(2008), 12961303.

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    Maier, K. Die Desarguessche Konfiguration. Deutsche Math. 4 (1939), 591641.

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    Olver, P.Classical Invariant Theory. London Mathematical Society Student Texts, No. 44. Cambridge University Press, 1999.

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    Pedoe, D.How many Pascal lines has a sixpoint? The Mathematical Gazette 25, 264 (1941), 110111.

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    Salmon, G. A Treatise on Conic Sections. Reprint of the 6th ed. by Chelsea Publishing Co., New York, 2005.

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    Seidenberg, A. Lectures in Projective Geometry. D. Van Nostrand Company, New York, 1962.

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    Sylvester, J. J. Note on the . . . six-valued function of six letters. In Collected Mathematical Papers, vol. II. Cambridge University Press, 1904–1912, pp. 264271.

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    Van der Waerden, B. L. Modern Algebra, vol. I. Springer, 1940.

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    Veronese, G. Nuovi teoremi sull’hexagrammum mysticum. Atti della Reale Accademie dei Lincei I (1877), 649703.

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