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In 1989, Lam, Thiel, and Swiercz used a massive case analysis on a Cray-1A to prove that no projective plane of this order exists, thus resolving the smallest case not covered by the Bruck–Ryser theorem. For 10 points each:
[10m] One, six, and what number are the three least positive integers n for which a finite field of order n does [emphasize] not exist?
ANSWER: ten (A finite field of order n exists if and only if n is a prime power. A projective plane of order n exists if a field of order n exists, but it is not known whether the converse is true.)
[10h] The existence of a projective plane of order n is equivalent to the existence of n minus one of these objects of order n. Euler theorized these objects in a problem about arranging 36 officers in a six-by-six grid.
ANSWER: mutually orthogonal Latin squares [or MOLSs; prompt on Graeco-Latin squares, Latin squares, or Euler squares]
[10e] The proof of Lam et al. relied on the theory of these objects that describe operations on binary strings. Golay and Hamming name binary examples of these objects.
ANSWER: codes [accept error-correcting codes, Golay codes, or Hamming codes]
<Arya Karthik, Other Science - Math>

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2023 Chicago Open08/05/2023Y106.0050%10%0%

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