[Guest]

Substitute each of the letters in this 3x3 square by a different digit from 0 to 9 such that the 8 sums constituted by the three rows, three columns and two main diagonals are consecutive integers (albeit, not necessarily in this order). Treat the rotation and reflection of a valid arrangement as the same solution.

A    B    C

D    E    F

G    H    I

Edited July 30, 2009 by K Sengupta

[Guest]

This solution works

2 9 6 next line 5 3 1 next line 4 7 8

everything adds up to 19. Seems like there should be another solution, any ideas?

[Guest]

That solution doesn't work.

Each diagonal,each row amd each column should yield different totals.

There are, I think, 3 different possible sets of integers for the totals:

{11, 12, 13,...18}

{12, 13,14,.... 19} or

{13, 14, 15,... 20}

However my trial and error approach so far hasn't found me any answers. I'm not ready to delve into an algebraic approach yet, though.

Donjar

[Guest]

Here is what I got so far and just like donjar trial and error with some logic included got me this far

6 7 1

5 2 8

3 4 9

Only missing 2 column and first column and last diagonal to complete the circle but I'm out of ideas

[Guest]

Something you might want to think about.

Out of the 8 totals, 4 must be odd and 4 even. The even rows/columns will have either 0 or 2 odd numbers and the odd rows/columns with have either 1 or 3 odd numbers.

Also, when the OP says 0-9 that means we can use '0'?

Edited July 30, 2009 by psychic_mind

[Guest]

Also, when the OP says 0-9 that means we can use '0'?

Yes, using '0' is permissible in terms of the given conditions.

[Guest]

i found 2 but i think there is one for every odd upper left corner

186

970

453

318

920

465

[Guest]

i found 2 but i think there is one for every odd upper left corner

186

970

453

318

920

465

The second one works but the first one is definitely not all consecutive integers.

[Pickett]

So...there are 2 possible solutions to this problem. Each with 8 different configurations/rotations...that gives a total possible 16 "answers" to this problem...but only 2 distinct solutions. The first can be found if you don't use the number "2" in the solution (so use numbers 013456789). The second can be found if you don't use the number "7".

The sums for those two are as follows:

for the solution that doesn't use the 2, the sums of the rows/columns/diagonals are 11, 12, 13, 14, 15, 16, 17, 18...and for the solution that doesn't use the 7, the sums are 9, 10, 11, 12, 13, 14, 15, 16.

1 8 6

9 7 0

4 3 5

and

3 1 8

9 2 0

4 6 5

[Guest]

Ups... I understood that you had to accomplish that having some kind of order.

I tried to do

1st row = x and then go down the matrix. In other words:

a + b + c = x

d + e + f = x+1

g + h + i = x+2

a + e + i = x+3

c + f + i = x+4

b + e + h = x+5

a + d + g = x+6

g + e + c = x+7

I guess I complicated things for myself

Would be nice to achieve this though and I will still try to do so...

I was doing some thinking and there are more than 85230 combination possible and if someone please give me a hint of how can we do this without trial and error

Thanks

[Guest]

It just shows that reading the question v e r y carefully is necessary. I missed the 0 to 9, and just assumed it was a conventional 1 to 9 square!!!!!

Donjar

[Quantum.Mechanic]

Ups... I understood that you had to accomplish that having some kind of order.

I tried to do

1st row = x and then go down the matrix. In other words:

a + b + c = x

d + e + f = x+1

g + h + i = x+2

a + e + i = x+3

c + f + i = x+4

b + e + h = x+5

a + d + g = x+6

g + e + c = x+7

I guess I complicated things for myself

Would be nice to achieve this though and I will still try to do so...

I was doing some thinking and there are more than 85230 combination possible and if someone please give me a hint of how can we do this without trial and error

Thanks

What if a + b + c isn't the smallest sum?

To brute force it, you need to generate the sums for each permutation, count the unique ones (8), and check that max-min = 7.

It only takes a minute or two on my machine to generate compute all of these, including the rotations and reflections. (It's debatable whether it takes longer to filter out the rotations and reflections, or just compute them all and filter the results.)