Anti-magic squares

[Joshua Zucker]

Hi folks,
I've got some kids who are middle school age but need some practice
with very basic arithmetic, like adding one-digit numbers. So I try
to find interesting and fun problems that will make them practice that
over and over and over again while they work toward some bigger goal.

One that I ran into recently is the idea of "anti-magic squares",
which probably have a better or more official name out there
somewhere. The idea is that instead of making the sums of all the
rows, columns, and two major diagonals equal (like you do for a magic
square), you make them all different. There's basically only one 3x3
magic square, but there are a lot of anti-magic ones! So I'm trying
to come up with some questions to make the exploration more
interesting. Here are my ideas so far -- any suggestions about what
order to do them in, or about some questions to add to this list?

1. Make an anti-magic square.

2. Compare your square with those of other classmates. Do you have
the same answer? If they are different, why are they different?

3. What numbers can be in the center of an anti-magic square?

4. Find a way to relate any anti-magic square with a 3 in the center
with an anti-magic square with a 7 in the center. Can you explain a
rule for this? Use your rule to explain why there are equal numbers
of anti-magic squares with 3 in the center and with 7 in the center.

5. Are there any 2x2 anti-magic squares? Why or why not?

Thanks,
--Joshua Zucker

[Japheth Wood]

Hi Josh,

Last fall, my MAT students and I ran a math festival at a public school in the South Bronx. Each MAT student selected, developed and implemented an engaging math activity that could be explored in about 10 minutes (the middle school students rotated stations).

One very interesting activity was similar to your anti-magic square activity: inspired by asking students to find a 3x3 magic square, the MAT student introduced intermediate goals - points for how many of the totals matched. The middle school students 3x3 grid was scored as follows:

All 8 totals match: 1000 points!

7 totals match: 500 points

6 totals match: 400 points
5 totals match: 300 points
4 totals match: 200 points
3 totals match: 100 points
2 totals match:  50 points

So, an anti-magic square would score 0 points.

This was a very successful 10 minute activity.

But, to add to your question list, how about 

6. What is the largest possible sum for any row, column or diagonal?

7. What is the least possible sum for any row, column or diagonal?

8. What is the average of the sums of your rows, columns and diagonals?

9. What is the smallest possible average? What is the largest possible average?

These are worth further exploration.

Japheth

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[Joshua Zucker]

On Sun, Jun 19, 2011 at 3:41 PM, Japheth Wood <jw...@bard.edu> wrote:
> 6. What is the largest possible sum for any row, column or diagonal?
> 7. What is the least possible sum for any row, column or diagonal?

Good warm-up questions, thanks!

> 8. What is the average of the sums of your rows, columns and diagonals?
> 9. What is the smallest possible average? What is the largest possible
> average?

This is a great idea! I can see how to figure out the answer to this
last question for an arbitrary placement of 1 through 9, but I don't
see anything but brute force for answering it when you're restricted
to anti-magic squares.

--Joshua Zucker

[Japheth Wood]

For #9, I suspect that most arrangements of 1-9 in a 3x3 grid are anti-magic squares. So a pigeonhole style argument could give a lower bound on the minimum average sum, and then brute force could show it's attainable. At least, that's what I think could happen (but I haven't tried to with out the details).
Something very similar happened with the Rubik's cube - theoretical arguments gave bounds on "God's number", and decades of brute force and programming verified it eventually last summer.
Japheth

[Joshua Zucker]

On Sun, Jun 19, 2011 at 6:45 PM, Japheth Wood <japhe...@gmail.com> wrote:
> For #9, I suspect that most arrangements of 1-9 in a 3x3 grid are anti-magic
> squares. So a pigeonhole style argument could give a lower bound on the
> minimum average sum, and then brute force could show it's attainable. At
> least, that's what I think could happen (but I haven't tried to with out the
> details).

That could be true but I think the fraction of anti-magic is smaller
than you might have expected. I wrote a quick computer program to try
all 9! arrangements (ignoring symmetry and so on, really brute force
in the silliest possible way; I wrote it quickly but it didn't run
quickly!) and it found 24960 of the 362880 were anti-magic.

Assuming all my brute force is correct, I get 107/8 for the minimum
possible average of the eight sums and thus 133/8 for the max because
of the symmetry around the number 5. My program gave both those
results so the fact that it checks with the symmetry is at least
somewhat reassuring. I also looked at a few of them by hand and
checked that they were antimagic.

By the way, I was surprised to find that of the 24960, there were
fully 7520 of them where none of the sums equalled 15. I thought that
avoiding 15 was trickier than that! My brute force also checked
something I had been too lazy to work out by hand, which is that there
are anti-magic squares with whatever number you want in the middle
spot.

--Joshua