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Franklin Square

Observations and Proofs:
Columns, General Observations

by Neil Abrahams

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COLUMNS, GENERAL OBSERVATIONS

  1. All columns are magical.

    Without loss of generality, notice from the construction algorithm that - in moving rightward column-by-column across the square - at each move, with the exception of the seed or end cells, values increase and decrease, respectively, in the n/2 plus and the n/2 minus rows. The values change in the same way when moving into either of the seed cell columns, with the exceptions noted above. Consequently, since each row has both a seed and an end cell and there are an equal number of plus and minus rows, n2 is added to what would otherwise be the values of as many cells in each of the seed columns as it is subtracted from. Thus, we can neglect the influence of the seed cell columns adjustment in this proof.

    Reconsider now what happens to the values of cells when moving, column-by- column, across a square. First note that the value of the cell in a row will either be increased or decreased by either the first or second element of that row's root pair.

    Without loss of generality, we will assume that the value is changed by the second element. This is 2x - 1, where x is the seed cell value for the row. Observe from the construction algorithm that - for exactly half of the rows - the (n/4 + 1)st. to the (3n/4)th rows, inclusive, the seed cells increase in value by one with each subsequent row. This means that the sum of the x values in the minus rows exceeds that of theplus rows in one half of the square by (n/2)/2 = n/4. From this, we can infer that the net value of the cells in this half of the square decreases by 2(n/4) = n/2 when moving rightward from one column to the next, since (2z - 1) - [2(n/4 + z) - 1] = - 2(n/4) = - n/2, where z is the sum of the value of the plus rows in the central half of the square.

    The opposite situation prevails in the remaining half of the square's rows. Thus, the net effect on the value of a column's cells when moving across the square is nil.

    Finally, since the value of the seed column pair must be n(n2 + 1), (Observation 3 and the construction algorithm), the value of each of its columns (as well as that of every other column) must be (n/2)(n2 + 1). Thus, every column is magical.

  2. The mean value of the cells in each column is (n^2 + 1)/2.

    This follows directly from Observation 11 and the fact that each column has n cells.

  3. Every order-2 subsquare (core) is (4/n)-magical.

    If the core is located in the seed columns, then its value is obviously 2(n2 + 1) = (4/n) *(n/2)(n2 + 1), or (n/4)th of the magic constant, so assume that it is located elsewhere. Without loss of generality, assume that its two rightmost cells are an even positive number of cells, say x cells, from their rows' left seed columns cells. Then the core's leftmost cells are x - 2 cells from their rows' right seed cells. Hence, recalling that the difference in value of every other cell is 2n, with the exception that n2 is added to (in a minus row) or subtracted from (in a plus row) what would otherwise be the value of the right seed cell (Observation 2), we see that the value of the core's lower row is:

    n2 + 1 + (x/2)(2n) + ([x - 2]/2)(2n) - n2 (if it is in an addition row), or
    n2 + 1 - ((x/2)(2n) - ([x - 2]/2)(2n) + n2 (if it is in a subtraction row).

    Since the core's two rows are adjacent, they must be of different types. Consequently, if one of the above values is that of the core's lower row, the other must be that of its upper row (and vice-versa). A similar argument can be made if the core's leftmost cells are an even positive distance from their rows' left seed cell. Hence, the value of a core given the above conditions is the sum of (1) and (2), or 2(n2 + 1), the same as that of a core located on the seed columns. Consequently, a core is (4/n)-magical.

  4. If a square's order is an even power of 2, it contains magical minisquares (but not half-magical ones); if its order is an odd power of 2, it contains half-magical minisquares, (but not magical ones).

    Recalling that the order of a Franklin square is 2p where p > 2, let o be the order of a square and p the power of 2 it equals . If o is a square number, then p is even, and minisquares of o, but not of o/2 cells, exist. If o is not a square number, then p is odd, and minisquares of o/2, but not of o cells, exist. Since any minisquare of n or n/2 cells, respectively, can be decomposed into n/4 or (n/2)/4 = n/8 cores, each of which is (4/n)-magical, any minisquare of n or n/2 cells, respectively, is magical or half- magical. Thus, if a square's order is an even power of 2, it contains magical minisquares (but not half-magical ones); if its order is an odd power of 2, it contains half-magical minisquares (but not magical ones).

  5. (i.) Cell pairs in adjacent columns and alternating rows are equivalent.

    (ii.) Cell pairs in adjacent rows and alternating columns are equivalent. (i.) Imagine two overlapping cores, A and B, with A's lower row being B's upper.

    Since all cores are equivalent and a number is equivalent to itself, we can infer that the upper cell pair of A and lower cell pair of B are also equivalent. As this can occur anywhere on the square, any pair of adjacent cells must be equivalent to all same-column cell pairs in alternating rows. Thus, all cell pairs in adjacent columns and alternating rows are equivalent.

    (ii.) The same reasoning applies to alternating column same-adjacent-row cell pairs.

    Thus, all adjacent rows cell pairs in alternating columns are equivalent.

  6. Every congruent quadruple of cells is half magical.

    On the square table below, consider the sixteen innermost cells (Region 1). From Observation 14 we know that cell pairs 28,36 and 30,38 and cell pairs 27,35 and 29,37 are equivalent. Similarly, cell pairs 20,21 and 36,37 and 28,29 and 44,45 are also equivalent. Since cell pairs 28,36 and 29,37 form a core, we can infer from the first set of equivalencies that cell pairs 30,38 and 27,35 are together equivalent to one. Similarly, we can deduce that cell pairs 20,21 and 44,45 are also equivalent to a core. Thus Region 1 - which must, from its size, be equivalent to four cores - has, less its corner cells, a value equivalent to three. Thus, its corner cells are equivalent to a core. Now add to this region the remaining lined cells. This new region is equivalent to nine cores. The cells without horizontal lines in them can be decomposed into groups equivalent to cores, as shown. Imagine removing these and the clear cells from the grid and bringing the remaining (horizontally lined) cells together. Look familiar? Your observation and the preceding reasoning let you infer that the corner cells of the non-clear region are also equivalent to a core. We can generalize this result to any even-ordered square region. Thus, every quadruple of cells is (n/4)-magical.

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Next Page: Observations on Columns & General Observations, cont.



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