# COMPOSITE SQUARES

Mathematically a composite number is an integer that is a combination of two or more primes. By that definition, the squares discussed in Order-2p Squares and Order-3p Squares and the powers of primes are composites. This section will concentrate only on composites of more than one prime, not the powers of primes. The discussion will be split into even and odd composites, as they must be treated differently.

## EVEN ORDER COMPOSITE SQUARES

It has been proven that squares of singly even order cannot be pan-magic. All squares of doubly even order can be pan-magic. Thus, the order-12 pan-magic squares are the smallest that will be considered here. I initially became interested in the order-12 squares during a discussion with Aale de Winkel about the implications of the compact term. I thought I had proof that {2compact4} meant that the figure was also {2compact2}. He pointed out that the proof did not work for order-12 squares. My proof was confined to order-2p figures. A further discussion of this is given below.

Order-12 base squares must be either binary or ternary. The order-12 pan-magic squares are constructed from 4 binary base squares and 2 ternary base squares. This creates a number of construction challenges. For instance, the {2compact2} feature can be expressed in the binary base squares but not the ternary base squares. Thus, it cannot be expressed in an order-12 pan-magic square made from base lines. {2Compact2} order-12 pan-magic squares can be made in other ways. There is an example below. My son Neil put together a program to create order-12 pan-magic from base lines and the examples below are largely a result of that effort. The order of the base squares can be shuffled as with the other constructions. Base squares 5 and 6 are always ternary base squares and the others binary. The base square order can be shuffled in any order to give a new order-12 pan-magic square with the same properties. There must be no repeats in the base cube order, however.

Square 1 is {2compact7}. If it were to follow the pattern based on order-2p squares, it would also be expected to be but {2compact3} and complete. It is neither.

Square 2 is just pan-magic with no special feature except a feature my son calls quadplete. This is the sum of 16 numbers evenly spaced in a grid pattern is any 10x10 sub-square within the order-12 square. Every order-12 pan-magic square I have examined has this feature.

Square 3 is {2compact7} and complete.

Square 4 has both Franklin V and what I call Franklin VW bent diagonals. The VW bent diagonals as the letters imply go back and forth three times while crossing the square. A Franklin W type bent diagonal is not possible.

Square 5 is {2compact4} but it is not {2compact2 or 6}. This square also has both Franklin V and Franklin VW bent diagonals.

Square 6 has four inlaid order-3 magic squares. They have their upper left corners at positions (4,4), (10,4), (4,10), and (10,10).

Square 7 has sixteen order-3 magic squares in a grid arrangement. Notice that the 1 is not in the upper left corner because an order-3 square cannot have its 1 in a corner. The order-12 square is also {2compact7} and complete.

The numbers in square 8 are only slightly different from those in square 7 but it is {2compact4}, not {2compact7} and not complete.

## ODD ORDER COMPOSITE SQUARES

In general for any odd order-o where o is not divisible by 3, the two series x mod o and 2x mod o can always be used as base lines to make pan-magic base squares with 0 ≤ x < o. When o = 3 the two series are the same (one forward and one backward), thus a pan-magic square cannot be made. For larger odd order-o squares divisible by 3, the two series will always give incorrect diagonals in one direction. Squares that are orders that are singly divisible by 3 can be pan-magic. For instance, John R. Hendricks created an order-15 pan-magic square. It does not appear, however, that these squares can be created using base lines. Composites of two different primes ≥ 5 must be ≥ 35. Squares of these orders are relatively easy to make from base lines.

More generally when o is prime, the two series rx mod o and sx mod o will create base lines that always make pan-magic base squares with 0 ≤ x < o. For composite numbers not divisible by 3, the additional restriction that r and s must not be factors of o must be applied. Even with that restriction, there are many other combinations of r and s series that do not give pan-magic base squares but there are many that do.

Two order-o base squares must be combined by multiplying one base square by o and adding the second. This will not always yield a valid pan-magic square for squares of composite order but will always yield a pan-magic square for squares of prime order. Uniform integral distribution must be confirmed for the composite squares. I believe that the two base squares o(x mod o and 2x mod o) plus (2x mod o and x mod o) will always yield a pan-magic square where the series in parenthesis represent the row and column base lines needed to make the base squares.

There are four order-35 pan-magic squares shown on the 35x35 worksheet of the SquareLines Excel Spreadsheet available on the Downloads page. The first two master base line sets are squares made as described above. The second is associated, the first is not.

The third and fourth magic squares on the worksheet are made by multiplying an order-5 pan-magic square by an order-7 pan-magic square and vice versa. The same two associated pan-magic squares were used for both multiplications. Both order-35 squares are associated but the top row and left column are not master base lines. The squares cannot be made from master base lines like the order-25 multiplied squares. Multiplication of master base lines does not work unless both base lines are the same length.

## COMPACT IMPLICATIONS

 144 1 120 121 48 85 72 73 60 97 24 25 3 142 27 22 99 58 75 70 87 46 123 118 134 11 110 131 38 95 62 83 50 107 14 35 9 136 33 16 105 52 81 64 93 40 129 112 137 8 113 128 41 92 65 80 53 104 17 32 7 138 31 18 103 54 79 66 91 42 127 114 140 5 116 125 44 89 68 77 56 101 20 29 6 139 30 19 102 55 78 67 90 43 126 115 133 12 109 132 37 96 61 84 49 108 13 36 10 135 34 15 106 51 82 63 94 39 130 111 143 2 119 122 47 86 71 74 59 98 23 26 4 141 28 21 100 57 76 69 88 45 124 117

There is a proof in Magic Constant Groups that if a cube is {2compact2} it is also {2compact4} or any other even number. An order-12 example that is {2compact2,4,6} is shown at right. This proof is easily extended to figures of any number of dimensions and/or order. At the time that this was originally written I thought I had also proved that a figure that is {2compact4} is also {2compact2}. However, that proof only works for figures of order-2p.

Since the proof works with any number of dimensions, the easiest number of dimensions to work with is one or a line. A simplified proof thus follows: For an order-8 line call the values a b c d e f g h. If this line is {1compact2} then a+b = b+c = c+d = d+e = etc. Since b+c = c+d then b = d. Thus by substitution a+b = a+d. Proving that the line is also {1compact4}.

As an example, for the {2compact2,4,6} square at right, let a = 33+113 = 146, then b = 16+128 = 144, c = 105+41 = 146, and d = 52+92 = 144. Thus a+b = a+d as above. The four pairs of numbers could start at any point in the order-12 square with the same result.

For the converse if the above order-8 line is {1compact4} then a+d = b+e = c+f = d+g = e+h = f+a = g+b = h+c. The last three require wraparound. Since d+g = g+b then d = b. Thus by substitution a+d = a+b. Proving that the line is also {1compact2}.

 1 121 36 129 4 124 30 132 7 118 33 135 65 95 44 83 68 98 38 86 71 92 41 89 117 15 136 25 114 12 142 22 111 18 139 19 106 58 75 54 103 55 81 51 100 61 78 48 2 122 35 128 5 125 29 131 8 119 32 134 66 96 43 82 69 99 37 85 72 93 40 88 115 13 138 27 112 10 144 24 109 16 141 21 107 59 74 53 104 56 80 50 101 62 77 47 3 123 34 127 6 126 28 130 9 120 31 133 64 94 45 84 67 97 39 87 70 91 42 90 116 14 137 26 113 11 143 23 110 17 140 20 108 60 73 52 105 57 79 49 102 63 76 46

For an order-12 line, a b c d e f g h i j k l, {1compact2} still means that the line is also {1compact4}. It is also {1compact6} by the same logic. The converse, however, does not follow. If the line is {1compact4} then a+d = b+e = c+f = d+g = e+h = f+i = g+j = h+k = i+l = j+a = k+b = l+c. Since d+g = g+j then g = j. By substitution a+d = a+j but a and j are 4 apart which is still {1compact4}. No substitution will give a different level of compact, thus {1compact4} does not imply {1compact2}. Squares 5 and 8 in the Order-12 Pan-Magic Squares generator above illustrate this fact.

If the order-12 line is {1compact6}, however, it must also be {1compact2}. For a {1compact6} line, a+f = f+k = k+d = d+i = i+b. Therefore f = d and d = b and thus f = b and by substitution a+f = a+b.

Similarly if the order-12 line is {1compact3} then it must also be {1compact7} like the {1compact3,7} at right. To see this a+c = b+d = c+e = d+f = e+g = etc. Since c+e = e+g then c = g. Thus by substitution a+c = a+g. Proving that the line is also {1compact7}. The converse, however, does not follow, i.e. {1compact7} does not imply {1compact3}.

 8 1 6 65 72 67 116 109 114 101 108 103 3 5 7 70 68 66 111 113 115 106 104 102 4 9 2 69 64 71 112 117 110 105 100 107 125 118 123 92 99 94 17 10 15 56 63 58 120 122 124 97 95 93 12 14 16 61 59 57 121 126 119 96 91 98 13 18 11 60 55 62 29 36 31 44 37 42 137 144 139 80 73 78 34 32 30 39 41 43 142 140 138 75 77 79 33 28 35 40 45 38 141 136 143 76 81 74 128 135 130 89 82 87 20 27 22 53 46 51 133 131 129 84 86 88 25 23 21 48 50 52 132 127 134 85 90 83 24 19 26 49 54 47

An example in the {2compact3,7} square could be 15+122 = 137. Then c would be 25+128 = 153, e would be 12+125 = 137, and g would be 22+131 = 153. Thus a+c = a+g as above proving that {2compact3} implies {2compact7}. Squares 1, 3, and 7 in the Order-12 Pan-Magic Squares generator above illustrate the fact that {2compact7} does not imply {2compact3}.

The next order-12 pan-magic square at right was contributed by Aale De Winkel. It is {2Compact4,7} but neither {2compact2} nor {2compact3}. The square is also complete and has 16 inlaid order-3 magic squares.

Generalizing, for order-4n squares, assuming that the magic square is large enough and/or wraparound is used, a square that is {2compactx} is also {2compact(2(x-1)y}. This is also true for figures of any number of dimensions.

Generally a square that is {2compact(2(x-1)y} will also be {2compactx}. But if (2(x-1)y-1) is a factor of the squares order or is an odd multiple of that factor then {2compact(2(x-1)y} does not imply {2compactx}. For example, for an order-28 square, {2compact8 or 22} does not imply {2compact2}. But if the square is {2compact2} it must also be {2compact8,22}. Seven, (8-1), is a factor of the squares order, 28, and 21, (22-1) is 7 + order/2. (15-1) is also a factor of the order and thus {2compact15} does not imply {2compact3}, but {2compact7, 11, or 19} do imply {2compact3,7,11,15,19}