Humongous Magic Square

Here is one of the biggest magic squares around. It is 15-by-15 squares in all. It was originally published by a German mathematician in 1604. (I got it from Mathematics From the Birth of Numbers. This is not only a nifty magic square, but the explanation that follows it below gives you a general method for creating odd-sided magic squares.

8 121 24 137 40 153 56 169 72 185 88 201 104 217 120
135 23 136 39 152 55 168 71 184 87 200 103 216 119 7
22 150 38 70 54 167 70 183 86 199 102 6 118 6 134
149 37 165 53 166 69 182 214 198 101 214 21 5 133 21
36 164 52 180 68 181 84 197 100 213 116 4 132 20 148
179 51 179 67 195 83 196 99 212 115 3 35 19 147 35
50 194 66 194 82 210 98 211 114 2 130 18 146 34 162
193 65 193 225 209 97 225 17 1 129 17 49 33 161 49
64 192 80 208 96 224 112 15 128 16 144 32 160 48 176
191 79 207 14 223 111 14 31 30 143 31 63 47 175 63
78 206 94 222 110 13 126 29 142 45 158 46 174 62 190
221 93 221 109 12 125 28 141 44 157 60 77 61 189 77
92 220 108 11 124 27 140 43 156 59 172 75 188 76 204
10 107 10 42 26 139 42 74 58 171 74 91 90 203 91
106 9 122 25 138 41 154 57 170 73 186 89 202 105 218

Did you find the pattern? Sort of? That's okay, it isn't completely clear until you're given the secret. So how does it work?

You begin by putting a 1 in the square that is just to the right of the center square (which is here the 8th row and 9th column). Then you enter increasing number, moving diagonally upward to the right. In this particular square you go from 1 to 7. Any time you reach the right side of the magic square is continued on the lefthand side of the magic square, one row up. So, in this case, since the 7 hits the last column in the second row, we put the 8 on row one starting in the far left column.

When you come to the top, or first, row, then you continue in on the bottom row of the next column. In this case, 8 happens to be in the top row in the first column, so we have put 9 in the bottom, or fifteenth, row of the next column (which is the second column in this case). Then you continue increasing your numbers diagonally up toward the right like before.

When you come to a square that already has a number in it, then you skip over that number and restart on the same row but two columns over. This happens when 15 runs into 1 in this magic square. Look and see where 16 in comparison to 15 in the magic square.

One special case in this magic square happens when 120 hits the upper right hand corner. Normally it would go in the bottom square of the far left column (the bottom left corner). But it is blocked by 106. So the two-step change gets made by putting 121 in the top square of the next column. Then, because the numbering has now reached the top row, it has to be continued on the bottom row of the next column, which is why 122 is in the third column of the bottom row.

You continue in this way until you get to the last space, and enter 225 just to the left of 113, which is in the center of the whole magic square.

Oh, by the way, did you figure out what every row, column, and diagonal adds up to? You can add one or two of them up if you wish, or you can do the following: Figure out the total number of squares (15 squared, actually, which = 15 X 15 = 225). Then multiply the total number of squares (225) by the total number of squares plus one (226) and divide the answer by 2. That equals (225 X 226) / 2 = 26,425. That's what the total of all of the squares is. Since there are 15 squares in each of the columns or rows and diagonals, you divide 26,425 by 15 and that equals 1,695, which is exactly what each and every row, column, and diagonal adds up to.

Now that's magic!