[Guest]

N is a 3x3 magic square which is constituted by using each of the digits from 1 to 9 exactly once.

Determine the probability that, reading from left to right, the sum of the first digit minus the second digit plus the third digit in each row, each column, and each main diagonal of N is the same.

Edited October 7, 2009 by K Sengupta

[Guest]

CORRIGENDUM

The original post was erroneous, and the corrected version is as follows:

"N is a 3x3 magic square which is constituted by using each of the digits from 1 to 9 exactly once.

Determine the probability that, reading from left to right, the first digit minus the second digit plus the third digit in each row, each column, and each main diagonal of N is the same."

Edited October 7, 2009 by K Sengupta

[Guest]

zero

[Guest]

I believe the only solution is a rotation/flip of

2 3 6

1 5 9

4 7 8

so the is a total of 8 solutions (think of 2 possible solution for each corner number in the upper left corner)

the total number of permutations is 9!=362880 so I get 1 in 45360

[Guest]

As there are 362,880 possible combinations (9!) and (so far) I've

found two (2) solutions, I posit that the probability is 2/362880;

or 1/181440 btw: the two solutions (I found) are:

8-9-6

7-5-3

4-1-2

4-1-2

7-5-3

8-9-6

-psyclist

Edited October 7, 2009 by psyclist

[superprismatic]

0. I agree with tbrophy. The squares in posts by psyclist and Doctor Moshe are not magic squares.

[Guest]

ok, after some additional thought;

There are 362,880 possible combinations (9!).

The solution has 4 mirrors (the two diagonals,

& the vertical and horizontal planes) that it can

"see" itself in that doesn't alter the same answer.

These "mirrors" derive 24 different solutions

(4!). Therefore the answer is 24/362880, or

1/15120

-psyclist

Edited October 7, 2009 by psyclist

[Guest]

sp - I guess I assumed that KS was using a generic definition of "magic square" to mean a square with all rows, cols and diags to result in the same answer and then defining the +,-,+ rule instead of a simple +,+,+ rule.

[Guest]

psychlist, I still stand by my answer. I contend your "mirrors" result in identical sol'ns and there are only 8 true variations

[Guest]

My good Doctor (with all respect), I must disagree in that

any combination of numbers where the positions of the digits

are different from any previous incarnation can only result

in a "new" solution. As I can craft 24 different possible

combinations - that, in and of itself mandates that there

are 24 different "solutions". Variations count as long as

they're unique - as these 4 mirrors prove.

-psyclist

[superprismatic]

sp - I guess I assumed that KS was using a generic definition of "magic square" to mean a square with all rows, cols and diags to result in the same answer and then defining the +,-,+ rule instead of a simple +,+,+ rule.

Oh, now I know how you interpreted the OP. I read it as the +,+,+ rule has to work as well as the +,-,+ one. I guess only KS knows for sure what he meant! Please enlighten us, KS.

[Guest]

Mea culpa!! After even more additional thought,

I now see that my 4 "mirrors" should cause the

results to double (hence the "mirrored" effect)

and NOT factorial-ized.... so there are indeed

8 unique results (4x2) and not 24 (4!) as I

had (erroneously) previously submitted.

Thank-you - good Doctor, for I wouldn't have

taken another look without your challenge!

Most humbly enlightened,

-psyclist

[Guest]

My good Doctor (with all respect), I must disagree in that

any combination of numbers where the positions of the digits

are different from any previous incarnation can only result

in a "new" solution. As I can craft 24 different possible

combinations - that, in and of itself mandates that there

are 24 different "solutions". Variations count as long as

they're unique - as these 4 mirrors prove.

-psyclist

I'm sorry but I don't see how you can get 24 unique soln's. It appears that you are saying there are 4 'mirrors' and 4 different starting positions or something like that and then throwing the factorial at it to get 4!.

Here is my reasoning: the 4 corner numbers must be corner number for all solutions. So there are 4 different possibilities for the number in the top left corner. For each of those possibilities the solution could be flipped 2 ways (i.e. transposed) for a total of 8 combinations as follows:

8-9-6 8-7-4

7-5-3 9-5-1

4-1-2 6-3-2

6-9-8 6-3-2

3-5-7 9-5-1

2-1-4 8-7-4

4-1-2 4-7-8

7-5-3 1-5-9

8-9-6 2-3-6

2-3-6 2-1-4

1-5-9 3-5-7

4-7-8 6-9-8

As for mirrors and rotation you have

orig up-left reflection line

vertical reflection line rotate CCW

horizontal reflection line rotate CW

up-right reflection line rotate 180

[Guest]

sorry, I was so busy writing up all those options I didn't see your reply.

glad we are on the same page

[Guest]

Oh, now I know how you interpreted the OP. I read it as the +,+,+ rule has to work as well as the +,-,+ one. I guess only KS knows for sure what he meant! Please enlighten us, KS.

I apologize for not replying earlier, as I was away for the last seven hours.

This problem is a modification of an exercise which appeared in the puzzle column of an old issue of a US based periodical.

Rereading the original puzzle, I now understand that the term “magic square” in that problem meant +,-,+ rule rather than being in conformity with the generic +, +, + rule.

Accordingly, the lacuna corresponding to the appearance of the words “3x3 magic square” (rather than just “3x3 square”) in the first paragraph of the current problem text stemmed from the said misinterpretation.