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Exercise 1.4. (a) Are P and Q orthogonal to each other? (b) Is P orthogonal to the Latin squares in (5)? Is Q? (c) Using the same alphabet, can you find another Latin square, R, for which P, Q and R are MOLS? At this point you may have conjectured that there are at most three MOLS of order 4. We shall now show that the largest number of MOLS of order n is n 1. Suppose we have a set of MOLS of order n. We can assume that the first row of each is arranged in ascending order, 0, 1, 2, 3, 4, ...,n 1, since if it were not we could simply rename the elements so that it fits this arrangement. Consider the elements in the first column of the second row in these squares. These elements must be distinct and different from 0. Why? They must be distinct since the ordered pairs (0,0), (1, 1), ..., (n 1, n 1) are produced in the first row when overlapping any two of these squares. They must be different from 0 since there can only be one 0 in the first column. Therefore we only have n - 1 possibilities, and we can have at most n - 1 MOLS of order n. If we have a set of n - 1 MOLS of order n then we say that it is a complete set of MOLS of order n. Here is the principal question which has generated over 200 years of research and remains largely unsolved. Fundamental Question: For which n are there complete sets of MOLS? This is another example of a question which is easily stated, can be explained to almost anyone, and has withstood the attempts of a great number of mathematicians for hundreds of years. It is precisely because of this that Gary Mullen, a well-known mathematician in this field, recently suggested it as the next Fermat problem. As we will see, this question is equivalent to one of the most important questions of finite geometry, namely when do projective and affine planes exist. There are some simple constructions for a complete set of MOLS of prime order which we shall now describe. First we recall some basic definitions of modular arithmetic. We say that a = b (mod n) if a b is a multiple of n. We can take as the representatives of all natural numbers modulo n the set {0, 1, 2, ...,n 1}, and use this set as our alphabet. When we add or multiply a and b we take the representative from this set that is equivalent to the sum or product modulo n. For example, 3+5 = 1 (mod 7) and 3(4) = 5 (mod 7). Using these operations, we can construct tables not unlike elementary school multiplication tables, but utilizing modular arithmetic. As an example, addition and multiplication tables for modulo 3 arithmetic are shown below. Notice that each cell in the grid is the result of its row heading and column heading under the specified operation modulo 3. + 0 1 2 0 1 2 * 0 0 1 2 0 0 0 0 1 1 2 0 1 0 1 2 2 2 0 1 2 0 2 1 Exercise 1.4. (a) Are P and Q orthogonal to each other? (b) Is P orthogonal to the Latin squares in (5)? Is Q? (c) Using the same alphabet, can you find another Latin square, R, for which P, Q and R are MOLS? At this point you may have conjectured that there are at most three MOLS of order 4. We shall now show that the largest number of MOLS of order n is n 1. Suppose we have a set of MOLS of order n. We can assume that the first row of each is arranged in ascending order, 0, 1, 2, 3, 4, ...,n 1, since if it were not we could simply rename the elements so that it fits this arrangement. Consider the elements in the first column of the second row in these squares. These elements must be distinct and different from 0. Why? They must be distinct since the ordered pairs (0,0), (1, 1), ..., (n 1, n 1) are produced in the first row when overlapping any two of these squares. They must be different from 0 since there can only be one 0 in the first column. Therefore we only have n - 1 possibilities, and we can have at most n - 1 MOLS of order n. If we have a set of n - 1 MOLS of order n then we say that it is a complete set of MOLS of order n. Here is the principal question which has generated over 200 years of research and remains largely unsolved. Fundamental Question: For which n are there complete sets of MOLS? This is another example of a question which is easily stated, can be explained to almost anyone, and has withstood the attempts of a great number of mathematicians for hundreds of years. It is precisely because of this that Gary Mullen, a well-known mathematician in this field, recently suggested it as the next Fermat problem. As we will see, this question is equivalent to one of the most important questions of finite geometry, namely when do projective and affine planes exist. There are some simple constructions for a complete set of MOLS of prime order which we shall now describe. First we recall some basic definitions of modular arithmetic. We say that a = b (mod n) if a b is a multiple of n. We can take as the representatives of all natural numbers modulo n the set {0, 1, 2, ...,n 1}, and use this set as our alphabet. When we add or multiply a and b we take the representative from this set that is equivalent to the sum or product modulo n. For example, 3+5 = 1 (mod 7) and 3(4) = 5 (mod 7). Using these operations, we can construct tables not unlike elementary school multiplication tables, but utilizing modular arithmetic. As an example, addition and multiplication tables for modulo 3 arithmetic are shown below. Notice that each cell in the grid is the result of its row heading and column heading under the specified operation modulo 3. + 0 1 2 0 1 2 * 0 0 1 2 0 0 0 0 1 1 2 0 1 0 1 2 2 2 0 1 2 0 2 1