For any composite integer N=mn we can arrange the integers 1 to N in a rectangular array with m rows and n columns such that the sum of each individual column equals N(N+1)/(2n). Naturally if there are an even number of rows this is trivial, since we can simply list the numbers sequentially left-to-right on odd-numbered rows and right- to-left on even-numbered rows, such as the 2x3 table 1 2 3 6 5 4 However, with an odd number of rows (greater than 1) we need to use a slightly more subtle arrangement. We can also assign the numbers 1 through n (in order) to the first row, i.e., we set A(1,k) = k k = 1,2,..,n The second row is assigned the numbers n+1 through 2n, placed consecutively beginning in the column just to the right of center, proceding to the right-most column, and then continuing from the left-most column to the center. Thus we set / k + (n-1)/2 + n for k=1 through (n+1)/2 A(2,k) = ( \ k + (n-1)/2 for k = (n+3)/2 through n The third row contains the numbers 2n+1 through 3n. The number 2n+1 is placed in the middle column, and 2n+2 is placed in the right-most column. Then 2n+3 is placed just to the left of 2n+1, and then 2n+4 is placed just to the left of 2n+2, and so on, alternating until the row is filled. These assignments can be expressed as / 3n + 2 - 2k for k = 1 through (n+1)/2 A(3,k) = ( \ 4n + 2 - 2k for k = (n+3)/2 through n Notice that each column from k=1 to (n+1)/2 has the following sum over the first three rows A(1,k) + A(2,k) + A(3,k) = [k] + [n+(n-1)/2+k] + [3n + 2 - 2k] = (9n+3)/2 whereas for the columns from k=(n+3)/2 to n the sums have the form A(1,k) + A(2,k) + A(3,k) = [k] + [(n-1)/2 + k] + [4n + 2 - 2k] = (9n+3)/2 Thus every column sums to the same value, (9n+3)/2, over these first three rows. This leaves an even number of rows to be filled with the numbers 3n+1 to N, so these can be assigned consecutively in the usual pattern for pairs of rows. To illustrate this pattern, here are two tables for N=35: n=5,m=7 m=7,n=5 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 6 7 11 12 13 14 8 9 10 15 13 11 14 12 21 19 17 15 20 18 16 16 17 18 19 20 22 23 24 25 26 27 28 25 24 23 22 21 35 34 33 32 31 30 29 26 27 28 29 30 35 34 33 32 31 This raises some interesting questions. For example, is it always possible to construct a "primitive" m x n equal-column array, meaning that no subset of the rows form an equal-column array. We've seen that there are primitive EC arrays for 2 and 3 rows, but what about arrays with more than 3 rows?

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