Observations and Proofs:
Columns, General Observations, cont.
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- The range-determining (extreme value) cells in alternate columns of the
square - starting with either the seed columns or the first non-seed ones and working
outwards from them in both directions - occur in the same rows. Additionally, these rows
are those with the two lowest seed values (for the alternate columns which start
with the seed columns), and with the two highest seed values (for the alternate columns
which start with the first non-seed columns).
From the construction algorithm and Observation 3, we can infer that the extreme value
cells for the seed columns occur in the bottom two rows. Since, when moving away from
both seed columns in both directions, the value of the cells in each type of row either
increases or decreases by 2n between alternate columns, the relative position of the
extreme value cells in alternating columns, starting with the seed columns, remains the
same as long as n2 - 2 (the difference between the extreme value cells in the seed
columns) exceeds i(4n), where i is an integer - the amount by which that difference is
reduced between alternating columns. Solving this inequality and multiplying by 2
columns per move, we see that the difference can be reduced across
(1.) [(n/4) - 1] 2 = n/2 - 2
columns in each direction. To find the extreme value cells of the remaining
n - (n/2 - 2) + 2 = n/2 columns, where the + 2 represents the fact that the
seed columns were not included in (1.), recall from the construction algorithm and
Observation 4 that the value of the first non-seed cells in the plus direction of a row is
x + (2n - (2x - 1)) = 2n - x + 1; while that of the first non-seed cell in the
minus direction is [(n2 + 1) - x - [2n - (2x - 1)] = x + n2 - 2n, with x
between 1 and n, inclusive.
Thus the value of the first non-seed cell in the plus direction of a row is minimized and
the value of the analogous cell in the minus direction of a row is maximized when x is at
a maximum. Since the greatest difference between the cells in the first two non-seed
columns occurs when the value of one of the cells in each of them in the minus direction
of its row is maximized while another of the cells in each of them in the plus direction of
its row is minimized, the two rows in which x reaches its maximum values (those with
the highest value seed cells) will contain the extreme value cells for the first two
non-seed columns. Using the same reasoning as for the seed and alternating columns, we
can say that all alternate columns in the square moving outward from the first non-seed
columns have their extreme value cells in these same two rows. This proves the
|Row i [root
||Difference (value of column cell - value of V cell) [Top-row a + row with its leftmost cell an even distance from its seed cell]
||b3 + 2n
||b[(n/2 - 1) - 1) +
[(n/2 - 1) - 1]/2(2n)
||[(n/2)/2](-2n) [[n -
||b[n - (n/2 - 2)] - [[(n -
(n/2 - 2)] - 1)]]/2(2n)
|n - 4
|n - 3
||bn-3 - 2n
|n - 2
|n - 1
- Any on-its-side V-shaped group of n cells (V) centered upon the horizontal axis of the square is magic.
Since we want to prove that a V is magical when it is centered on the horizontal axis of
the 2-dimensional square, we make no assumptions about where it is centered. We start
our analysis with the column containing the open end of our V and work centerwards,
moving one level inward (from both ends) at a time. At each new level we will note the
magnitude and direction of the difference in cell value between the cell on the V and that
in the same-row end column. Since the end column is magical, if the overall difference in
value between the cells on the V and those in the column is nil, the V must be magical.
Note that we can neglect the adjustment in values of cells in the seed columns for the
same reason that we did above. Because we need to keep track of direction of change of
value as well as magnitude, we will assume, without loss of generality, that the top cell of
the V occurs in a plus row and in a column an even number of cells to the right of its
row's seed cell. Additionally, we will index root pairs by rows, with the root pair of the
ith row being (ai,bi) (see table).
We begin by noting that, since the two end cells are in the column, the difference in
question for the outermost rows (level 0) of the V is 0. At the first level, we see that the
value of the V cell in the upper row is b1 less than that of the same-row end column cell.
Meanwhile, the value of the V cell in the lower row exceeds that of the same-row end cell
by bn - 1. Continuing to the second level, we note that the relevant absolute differences
for both upper and lower rows are the same (Observation 1) but in opposite directions by
virtue of their types differing. Thus, the net effect on the V's value relative to that of the
end column at this (and all subsequent even-numbered levels) is nil. At the third, (and all
subsequent odd-numbered levels), the value of the V cell in the upper row is b3, (plus a
multiple of 2n), less than that of the relevant column cell while that of the lower row is
again bn - 3, (plus a multiple of 2n), greater than that of the relevant column cell.
From all of the foregoing, we can deduce that, if the top cell of a V is in a plus row, then
it will be magical if, and only if, the sum of the second elements of alternating rows -
starting with the two center rows and working outwards - on both sides of the V, are
equivalent. Since the sum of all root pairs is 2n, a constant, this equivalence occurs in the
same rows for the first element of root pairs as for the second. Since:
- we would have used the first element in our analysis where we had used the second
had we assumed that the top row was a minus row, and;
- the reasoning for a left-facing V is identical to that for a right-facing V
our result can be generalized to any on-its-side V. Since we are not making any
assumptions about where the V is centered, in order to see which on-their-side Vs are
magical, we need to think of our root pair list as circular; any two adjacent rows might be
considered the center. We then find which choice of two adjacent rows for the center will
cause the same element - we'll use the second because it's easiest to work - of every other
row's root pair, when moving outward n/2 - 1 rows from one row in it in
one direction, to equal the same sum when moving outwards from the center in the other
direction (see table).
- Any upright or upside-down V-shaped group of n cells (V) is magical if, and only if,
it is centered an odd number of cells away from the center of the seed column cells.
- taken together, the pairs of cells in every row which any V traverses are either congruent to or the same as the cells in the pair of half-columns in the V's center, and
- the pair can be decomposed into n/4 cores, each of which is (4/n)- magical - with one class of exceptions,
any V is magical.
For the exceptions, recall that some congruent cell pairs have values differing by n2. If a V has an even number of these 'abnormal' congruent pairs (see Reason 4), it will still be magical since half of them will be 'supernormal' and half will be 'subnormal' (since they must appear in alternating rows). This occurs whenever a V is centered on neither the seed columns nor a column pair centered an even number of cells from their center. In the remaining cases, the V will have an odd number of 'abnormal' congruent pairs and will not be magical. This proves the conjecture.