Finding Graeco-Latin Squares Curtis Bright, Maplesoft 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 A Latin square of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZ0aW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYrRjk= matrix containing integer entries between 1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn such that every row and every column contains each entry exactly once. Two Latin squares are orthogonal if the superposition of one over the other produces all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ== distinct pairs of integers between 1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. A pair of orthogonal Latin squares is called a Graeco-Latin square. They were termed that by the mathematican Leonhard Euler who used Latin characters to represent the entries in one of the squares and Greek characters to represent the entries in the other square. For visualization purposes the entries of Graeco-Latin squares may be represented by different colours. For example, the image above shows a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEjMTBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RKCZ0aW1lcztGJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJRitGL0Yy Graeco-Latin square found by Maple after 23 hours of computation time. The entries of the second Latin square (represented by the small squares in the image) are superimposed onto the entries of the first Latin square. Euler studied the orders LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for which Graeco-Latin squares exist and found methods for constructing them when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn was odd or a multiple of 4. Since such squares do not exist for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjJGJ0YyRjxGMkY8 and he was unable to find a solution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjZGJ0YyRjxGMkY8 he conjectured that Graeco-Latin squares do not exist when 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. The problem was not solved until 1960 when it was shown that Graeco-Latin squares exist for all orders except 2 and 6 by R. C. Bose, S. S. Shrikhande, and E. T. Parker. Using Maple we can automatically construct (or disprove the existence of) Graeco-Latin squares for small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn without any knowledge of search algorithms or construction methods. To do this, we encode the problem in Boolean logic and use Maple's built-in efficient SAT solver to perform the search. A SAT solver takes as input a formula in Boolean logic and returns an assignment to the variables which makes the formula true (if one exists). See the Satisfy command for more information.

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= denote the two matrices in an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJnRpbWVzO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTkYrLyUlc2l6ZUdRIzEyRidGMkY8 Graeco-Latin square. We will use the Boolean variables 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 for 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 to denote that the integer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJqRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZBth entry of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and similarly define the Boolean variables 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.

To use a SAT solver we need to encode constraints in Boolean logic that specify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJCRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB is a Graeco-Latin square. There are three kinds of constraints: * Those that specify that every entry of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= is an integer between 1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRj0vJSpsaW5lYnJlYWtHUSVhdXRvRictRiw2I1EhRicvJSVzaXplR1EjMTJGJ0YyL0Y2USdub3JtYWxGJw==. * Those that specify that every row and column of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= contain no duplicate entries. * Those that specify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= are orthogonal. Additionally, there are constraints which are not logically necessary but help cut down the search space.

We need to specify that all entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= are well-defined, i.e., consists of a single integer from 1 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. First, we generate constraints that say each entry contains at least one integer from 1 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. These have the form 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 for each 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 and similarly for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=: atLeastOneInteger := proc(n) local k, Aconstraints, Bconstraints; Aconstraints := Array(1..n, 1..n, (i, j) -> &or(seq(A[i, j, k], k = 1..n))); Bconstraints := Array(1..n, 1..n, (i, j) -> &or(seq(B[i, j, k], k = 1..n))); return entries(Aconstraints, nolist), entries(Bconstraints, nolist); end proc: Next, we generate constraints the say that each entry contains at most one integer from 1 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. These have the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiotRi82JlEiaUYnRjJGNUY4LUkjbW9HRiQ2LlEiLEYnRjUvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRi82JlEiakYnRjJGNUY4RkAtRi82JlEia0YnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GQTYuUSomSW1wbGllcztGJ0Y1RkRGRi9GSUY3L0ZLRjRGTEZORlBGUi9GVVEsMC4yNzc3Nzc4ZW1GJy9GWEZmby1GQTYuUSJ+RidGNUZERkZGY29GSkZMRk5GUEZSRlQvRlhGVi1GQTYuUSYmbm90O0YnRjVGREZGRmNvRkpGTEZORlBGUkZURmdvLUYsNiZGLi1GIzYqRj1GQEZaRkAtRi82JlEibEYnRjJGNUY4RjJGNUY4RmpuRl1vRjVGRA== (alternatively, 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) for each entry LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJqRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB and each distinct pair 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 and similarly for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. atMostOneInteger := proc(n) local k, l, Aconstraints, Bconstraints; Aconstraints := Array(1..n, 1..n, (i, j) -> seq(seq(A[i, j, k] &implies &not(A[i, j, l]), l=k+1..n), k=1..n)); Bconstraints := Array(1..n, 1..n, (i, j) -> seq(seq(B[i, j, k] &implies &not(B[i, j, l]), l=k+1..n), k=1..n)); return entries(Aconstraints, nolist), entries(Bconstraints, nolist); end proc:

We need to specify that the entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= are Latin squares, i.e., all columns and rows contain distinct entries. These have the form 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 where 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 but 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 is in the same row or column as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJqRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW5HRiQ2JVEiMUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEmJmxlcTtGJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnRjUtRk02JlEibkYnRlBGL0ZTLyUlc2l6ZUdRIzEyRidGL0Yy, and similarly for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. sameRowOrColumn := proc(i, j, n) local ip, jp, k, sameRow, sameCol; sameRow := {seq([i, jp], jp=1..n)} minus {[i, j]}; sameCol := {seq([ip, j], ip=1..n)} minus {[i, j]}; return sameRow union sameCol; end proc: distinctnessConstaints := proc(n) local i, j, k, p, ip, jp, constraints; constraints := Array(1..2, 1..n, 1..n, 1..n, 1..n); for i from 1 to n do for j from 1 to n do for p in sameRowOrColumn(i, j, n) do ip, jp := op(p); constraints[1, i, j, ip, jp] := seq(A[i, j, k] &implies &not(A[ip, jp, k]), k=1..n); constraints[2, i, j, ip, jp] := seq(B[i, j, k] &implies &not(B[ip, jp, k]), k=1..n); end do; end do; end do; return select(x->x<>0, [entries(constraints, nolist)])[]; end proc:

Finally, we need to specify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= are orthogonal, i.e., for every pair LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJsRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB there exists some pair LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJqRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB such that 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 holds. These constraints are of the form 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 for each pair LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJsRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB. orthogonalityConstraints := proc(n) local i, j, constraints; constraints := Array(1..n, 1..n, (k, l) -> &or(seq(seq(A[i, j, k] &and B[i, j, l], j=1..n), i=1..n))); return entries(constraints, nolist); end proc:

Although these constraints are not strictly necessary they shrink the search space and thereby make the search more efficient. In general when the search space splits into symmetric subspaces it is beneficial to add constraints that remove or \342\200\234break\342\200\235 the symmetry. Graeco-Latin squares LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJCRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZB have a number of symmetries, in particular, a row or column permutation simultaneously applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= produces another Graeco-Latin square. Also, any permutation of 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 may be applied to the entries of either LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. The result of these symmetries is that any Graeco-Latin square can be transformed into one where the first row and column of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= has entries in ascending order (by permuting rows/columns) and the first row of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= has entries in ascending order (by renaming the entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=). Thus, we can assume the constraints 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 (and similarly for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn) as well as 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. symmetryBreakingConstraints := proc(n) local i, j; return seq(A[1, j, j], j=1..n), seq(A[i, 1, i], i=2..n), seq(B[1, j, j], j=1..n); end proc:

We use the Satisfy command from the Logic package which finds a satisfying assignment of a logical formula if one exists. We use the RealTime command from the CodeTools package to measure how quickly the solution is found. Given an order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn we define the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlE2ZmluZEdyYWVjb0xhdGluU3F1YXJlRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGMi9GNlEnbm9ybWFsRic= that returns a satisfying assignment (or NULL if none exists) to the Graeco-Latin constraints in order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn as well as the time it took the SAT solver to find them. constraints := proc(n) return atLeastOneInteger(n), atMostOneInteger(n), distinctnessConstaints(n), orthogonalityConstraints(n), symmetryBreakingConstraints(n); end proc: findGraecoLatinSquare := proc(n) local allConstraints := &and(constraints(n)); return CodeTools:-RealTime(Logic:-Satisfy(allConstraints)): end proc:

We use commands from the plots and plottools packages to visually display the found Graeco-Latin squares using different coloured squares to represent the different entries. The entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= will be drawn as small squares inside the squares representing the entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=. If a satisfying assignment was found the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNigtSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJqRidGNEY3RjovJSVzaXplR1EjMTJGJ0Y3RkFGN0ZBRlpGN0ZBth entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= can be found by iterating through the satisfying assignment and recording the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic=s that make 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 and 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 true. The orders LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzEyRidGMi9GNlEnbm9ybWFsRic= up to 5 can be solved almost instantly and the orders up to 8 can be solved in around 30 seconds on a modern CPU. with(plots): with(plottools): colors := ColorTools:-GetPalette("Patchwork"): drawEntryA := proc(i, j, k) rectangle([j,-i],[j+1,-i+1],color=colors[k],style=polygon); end proc: drawEntryB := proc(i, j, k) rectangle([j+0.25,-i+0.25],[j+0.75,-i+0.75],color=colors[k],style=polygon); end proc: for n from 1 to 8 do result := findGraecoLatinSquare(n); if whattype(result) = float then printf("Found that no Graeco-Latin squares of order %d exist in %.2f seconds.\134n", n, result); else computationTime, satisfyingAssignment := result; entriesA := Array(1..n, 1..n); entriesB := Array(1..n, 1..n); for eq in satisfyingAssignment do if rhs(eq) and op(0, lhs(eq)) = A then entriesA[op(1..2, lhs(eq))] := op(3, lhs(eq)); elif rhs(eq) and op(0, lhs(eq)) = B then entriesB[op(1..2, lhs(eq))] := op(3, lhs(eq)); end if; end do; printf("Found a Graeco-Latin square of order %d in %.2f seconds:\134n", n, computationTime); print(display(seq(seq(drawEntryB(i, j, entriesB[i, j]), j=1..n), i=1..n), seq(seq(drawEntryA(i, j, entriesA[i, j]), j=1..n), i=1..n), scaling=constrained, axes=none, size=[50*n,50*n])); end if; end do:

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