Page 4

Franklin Square

Observations and Proofs: Rows

by Neil Abrahams

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ROWS:

  1. The increase in value, starting with the seed cell, in the plus direction between every other cell in a row is 2n. This result follows directly from the fact that we construct each row by alternately adding, starting with the row's seed cell, the two elements of the root pair for that row - the sum of which is 2n - to the value of each cell to get that of the next one.

  2. The values of the n2 cells of an order-n square include every integer from 1 to n2, with each occurring exactly once. This can be seen by converting the construction algorithm, less any positioning information, into a function which generates the cell values of a square of a given order. Below is a Mathematica file which does this for an order-4 square, were it to exist.

    Clear[f,i,n,x]; f[i_] := f[i - 1] + If[Mod[i,2] == 0,(2n - (2x - 1)),(2x - 1)]; f[1] = x;

    [Define a recursive function f to generate cell values for any row (in the positive direction) of an order-n square with the variable x as its seed value. The ith cell's value will be that of the (i - 1)st plus either: (1) 2n - (2x - 1) - if it is an even distance from the seed cell - or (2) 2x - 1 if it is not. Then set the value of the initial (seed) cell of the row at x.]

    Table[Table[f[i],{i,4}],{x,4}] [Generate a list of lists of the values of the ith cell of each row (with i ranging from 1 to 4). The outer list (of xs, the rows' seed cell values) also ranges from 1 to 4. (n remains the order of the square.)]

    {{1, 2 n, 1 + 2 n, 4 n},
    -1 + 2 n, 2 + 2 n, -1 + 4 n},
    -2 + 2 n, 3 + 2 n, -2 + 4 n},
    -3 + 2 n, 4 + 2 n, -3 + 4 n}}

    Union[Flatten[%] /. n-> 4] [Replace n with 4, combine the lists, and put the new list's elements into standard numerical order.]
    {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

    You can see that each value from 1 to n2 = 16 appears only once in the final result. A proof would require showing that the same result would occur for all acceptable values of n. You can see this would occur by noting the pattern within and between the sets in the first output and extending this pattern to what it would be were there n elements in each of n sets. The first elements of the n sets would comprise the integers from 1 to n. The next n integers - from -(n - 1) + 2n = n + 1 to 2n - would be accounted for by the sets' second elements, starting with the last set and working up to the first.

    Next, the n integers from 1 + 2n to n + 2n = 3n would be culled from the sets' third elements, ending with the last set. Then it's back to the first using the fourth elements. This down and up cycle would occur n/2 times in a model representing a square of order-n. The final element would be the nth element in the first set - n*n = n2. After sorting, each integer from 1 to n2 would appear exactly once, proving the conjecture.

  3. The value of the seed column pair cells of each row is n2 + 1.

    Let the value of a row's seed cell be x, then, since:
    • the value of the next cell in the plus direction is the difference between 2n and 2x - 1, and
    • the value of every cell after that is 2n greater than the second cell before it, the value of the end (nth.) cell in that row is x + [2n - (2x - 1)] + 2n (n/2 - 1) = n2 + 1 - x.


    Consequently, the value of the seed column pair cells of each row is n2 + 1. Note that since if another cell followed the end cell its value would be (n2 + 1 - x) + (2x - 1) = n2 + x, each row's seed cell value is n2 less than what it would be were it a 'normal' cell.

    Viewing the same phenomena from the other direction, each row's end cell is n2 greater than what it would be were it a 'normal' cell.

  4. Every pair of congruent cells is either equivalent or differs in value by n2. All cell pairs congruent to a row's seed column's cell pair are equivalent to it. From the construction algorithm, we know that - with the exception of the seed column cells - the same value that is added to one of the cells of a pair of adjacent cells to get the value of its adjacent non-pair cell is subtracted from the value of the other cell to get the value of its adjacent non-pair cell. This is repeated at each level of movement away from the initial pair because there will always be an even number of cells between the members of each new pair of cells. From this it follows that, starting with an initial pair of adjacent cells and moving outward from them, until both the seed and the end cell for the row have been reached (assuming they are not the initial pair), the value of every congruent cell pair is the same as that of the initial one.

    There are two possibilities for the exceptions:
    • the seed cell isreached first, or;
    • the end cell is reached first.


    If it's the former, then the value of the end cell and subsequent congruent pairs will exceed that of the previous ones by n2; if the latter, then the value of the seed cell and subsequent pairs will be n2 less than that of the previous. If the seed and end cells of a row constitute the initial pair no change occurs, since - in moving outwards from them - neither the seed nor the end cell will be reached.

  5. The mean value of the range of all of the rows is n2 - n.

    Note, from Observation 3, that each row's range, or difference in value between end and seed cells, equals (n2 + 1 - x) - x = n2 + 1 - 2x, where x is the value of the seed cell. Since x takes on all values from 1 to n; the ranges of individual rows must vary from n2 - 1 (for the row with greatest range) to n2 + 1 - 2n (for the row with the lowest range) in decrements of 2. Since the mean of this set of values is n2 - n, the mean value of the range of all of the rows is n2 - n.

  6. All rows are magical.

    Since
    • the value of any row's seed columns cell pair is n2 + 1 (Observation 3), and
    • each of the other n/2 - 1 cell pairs in that row is congruent, and therefore equivalent to (Observation 4), each row's value must be (n/2)(n2 + 1), or the magic constant.


    Therefore, each row must be magical.

  7. The mean value of the cells in every row is (n2 + 1)/2. This follows directly from Observation 6 and the fact that there are n cells in every row.

  8. The median values of each row appear in the cell pair congruent to and farthest from the seed column's cell pair, or what might be called the row's mirror-seed column's cell pair.

    This follows directly from the construction algorithm and the definition of congruent cell pairs.

  9. (i.)The range of the bottom row equals that of the square itself (ii.) The median values of the square appear in the bottom row's mirror-seed column cell pair.

    (i.) This follows directly from the fact that the value of both the bottom row's and square's range is n2 - 1 (Observation 2 and Reason 5).

    (ii.) Note first, from the construction algorithm, that the distance from both the bottom row's seed cell to its right mirror-seed cell and from its end cell to its left mirror-seed cell is n/2. If we combine this with Observation 1 and the construction algorithm we can deduce that the value of the right cell of the mirror-seed cell pair for the bottom row must be [(n/2)/2](2n) = n2/2 greater than that of its row's seed cell, while the value of the other mirror-seed cell must be n2/2 less than that of its rows end cell. From the construction algorithm and Observation 3, we know that the values of the seed and end cells, respectively, of the bottom row of a square are 1 and n2. Consequently, the respective values of the right and left mirror-seed cells of the bottom row of a square must be 1 + n2/2 and n2 - n2/2 = n2/2. Since these are the median values of the consecutive integers from 1 to n2, and these are exactly the values of the cells on the square, (Observation 2), the median values of the square appear in the bottom row's mirror-seed column cell pair.

Next Page: Observations on Columns, General Observations



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