Another Easy Puzzle

[Alfie]

Arrange the numbers 1-169 into a 13x13 square where every row and column and both diagonals add to the same number.

I can do it in less than five minutes and without a calculator. (And have done so.)

[Amb]

Well I can do a 17 by 17? (An odd numbered square is very easy to create)

code:

---

155 174 193 212 231 250 269 288   1  20  39  58  77  96 115 134 153


173 192 211 230 249 268 287  17  19  38  57  76  95 114 133 152 154


191 210 229 248 267 286  16  18  37  56  75  94 113 132 151 170 172


209 228 247 266 285  15  34  36  55  74  93 112 131 150 169 171 190


227 246 265 284  14  33  35  54  73  92 111 130 149 168 187 189 208


245 264 283  13  32  51  53  72  91 110 129 148 167 186 188 207 226


263 282  12  31  50  52  71  90 109 128 147 166 185 204 206 225 244


281  11  30  49  68  70  89 108 127 146 165 184 203 205 224 243 262


 10  29  48  67  69  88 107 126 145 164 183 202 221 223 242 261 280


 28  47  66  85  87 106 125 144 163 182 201 220 222 241 260 279   9


 46  65  84  86 105 124 143 162 181 200 219 238 240 259 278   8  27


 64  83 102 104 123 142 161 180 199 218 237 239 258 277   7  26  45


 82 101 103 122 141 160 179 198 217 236 255 257 276   6  25  44  63


100 119 121 140 159 178 197 216 235 254 256 275   5  24  43  62  81


118 120 139 158 177 196 215 234 253 272 274   4  23  42  61  80  99


136 138 157 176 195 214 233 252 271 273   3  22  41  60  79  98 117


137 156 175 194 213 232 251 270 289   2  21  40  59  78  97 116 135

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[Amb]

The trick is to start with the number one in the middle square of the top row. The numbering follows upward diagonally. If the number reaches an edge then it simply wraps to the other side, and still up one. eg a 5 by 5

code:

---

..1..


.....


.....


....3


...2.

---

If the numbers be written ever collide trying to take a used square, then the number simply drops down one square and continues

code:

---

..18.


.57..


46...


....3


...2.

---

And finally becomes:

code:

---

17 24 .1 .8 15


23 .5 .7 14 16


.4 .6 13 20 22


10 12 19 21 .3


11 18 25 .2 .9

---

Now you go and do 13 X 13!


[This message has been edited by amb (edited 02-14-2000).]

[Alfie]

Just what I thought, way to easy for this group. Anyone want to try a 6x6? Bwahahahahaha.

[mithrandir]

code:

---

22 21 24 25 6  7 


20 23 27 26 5  4 


3  0  17 16 35 34 


1  2  19 18 33 32 


31 30 8  9  12 15 


28 29 10 11 14 13 

---

[This message has been edited by mithrandir (edited 02-12-2000).]

[Ghost Post]

There's a definite pattern in that one. 2-by-2 blocks, the blocks arranged similar to the the numbers in the 3-by-3 square. And a pattern within the 2-by-2 blocks.

Does it extend to 8-by-8? 10-by-10?

[Alfie]

How did you get that? I didn't know of an even order pattern.

[Amb]

What about a three dimensional magic "Square" (or now a magic cube). Is that even possible?

eg

code:

---

11 and 11


11     11  Works for a 2 by 2 cube, but the numbers are not sequential.  

---

I would be fascinated to know

[araya]

I know a general method for creating a nxn magic square where n is odd, or where n is divisible by 4 (different methods).

Magic cube: Richard Myers discovered a general method for finding an order-8 magic cube, when he was 16 years old.

[Amb]

Would you mind telling me the method for n where n mod 4 = 0. That would be interesting. (I have already explained the n mod 2 = 1 method in my higher posts)

Thanks Araya

[araya]

Sure, amb, I'll explain both methods since mine is different from yours

for odd order squares,

Place 1 in upper right corner. Move one cell to the left and 2 cells down and place 2. Continue moving in this way. If you land outside the square, wrap around to the opposite side (move up or right n cells as necessary). If you are going to land in an occupied square, instead move two cells to the left. Magic squares made in this way are also "pandiagonal" meaning if you tiled a plane with it, any nxn square in the plane is also magic.

code:

---

13 25 07 19 01


17 04 11 23 10


21 08 20 02 14


05 12 24 06 18 


09 16 03 15 22

---

Method for n mod 4 = 0 squares:

Divide the square into 4 subsquares, and draw the two main diagonals in all of these. Now, starting in the top left cell, insert numbers sequentially starting with 1 by moving left to right, only inserting a number if it is not crossed by a diagonal.

code:

---

xx 02 03 xx xx 06 07 xx


09 xx xx 12 13 xx xx 16


17 xx xx 20 21 xx xx 24


xx 26 27 xx xx 30 31 xx


xx 34 35 xx xx 38 39 xx


41 xx xx 44 45 xx xx 48


49 xx xx 52 53 xx xx 56


xx 58 59 xx xx 62 63 xx





Now, starting in the lower right, count up from 1 and fill in the empty cells.

---

source: Ancient Puzzles by Dominic Olivastro

[Amb]

Thanks araya, I love magic squares, but until today have never known a pattern for even squares but this pushes me closer and higher in what I know. Thanks

[Alfie]

Thanks. These make great puzzles for my pre-cal class at school.

[Amy]

Wow, that would make it an "Educator Idea"! I haven't seen one of those before...

[mathgrant]

I like this. I can't help but bump it.

[CzarJ]

Well, somebody had to say it...

Magic Hypercube?

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Basket-Weaving For Donuts, Where You Weave Baskets And Get Donuts.

[Courk]

I love threads where I learn something!

Thankyou!!!

[CrystyB]

What is the length of the side, CzarJ? You made me develop a Pascal program for finding it, but you still need to provide the input data...

[CzarJ]

I don't know... I just want a Magic Hypercube... It'll be cool

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Basket-Weaving For Donuts, Where You Weave Baskets And Get Donuts.

[Ghost Post]

I realize this is probably a stupid question, and no I'm not a math teacher . What's a hypercube?

[CzarJ]

It's a cube in 4 dimensions, basically. I could try to give a more mathematical explanation, but it probably wouldn't work out that well...

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Basket-Weaving For Donuts, Where You Weave Baskets And Get Donuts.