Determine the number of ways of placing the numbers $1, 2, 3, \dots, 9$ in a circle, so that the sum of any three numbers in consecutive positions is divisible by $3.$ (Two arrangements are considered the same if one arrangement can be rotated to obtain the other.)

I've experimented with possible combinations and found that it works when we put a multiple of 3 next to a number one more than a multiple of three beside a number that is two more than a multiple of 3. If we continue with this pattern around the circle, it works.

However, I'm curious in finding a more systematic approach than listing out all different combinations.

Thanks in advance!

  • $\begingroup$ Well consider the numbers in position $k$ and $k+3$ will have to be congruent $\mod 3$.. $\endgroup$ – fleablood Jul 8 at 5:08

In general, suppose we have the numbers $1,2,\dots,3n$, and we would like to place them in a circle so that the sum of any three consecutive terms is divisible by $3$.

Observe that the numbers at positions $k$ and $k+3$ must always be congruent modulo $3$. Thus we can partition the points along the circle into three sets which stand for the residues modulo $3$ of the positions. If we fix the number $1$ at, say, position $1$, then this tells us that every point at position $3k+1$ has residue $1$ modulo $3$.

Now we have a choice: either the numbers at positions $3k+2$ have residue $0$, or they have residue $2$. Either way, note that each of the three "partition classes" can be arranged in $n!$ different ways, giving us $2(n!)^3$ possibilities.

In this particular case, $n=3$, and the answer is $432$ (if rotations are counted as the same).

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  • $\begingroup$ If rotations are the same we can assume position $1$ has $3$. Then positions $4$ and $9$ have $6$ and $9$. there are $2$ ways to do that. position $2$ is either $\pm 1\pmod 3$. There are $2$ choices. position $2\equiv position 5\equiv position 8$ so there are $3!$ ways to do that. And $position 3\equiv position 6\equiv \position 9$ so there are $3!$ ways to do that. So there are $2*2*6*6=144$ ways if rotations are counted the same. If reflections are counted the same there are $2*6*6=72$ ways. $\endgroup$ – fleablood Jul 8 at 5:37

Brain storming. If we label the number positions as $a_1,.....,a_9$ then $a_k+a_{k+1} + a_{k+2} \equiv 0 \equiv a_{k+1} + a_{k+2} + a_{k+3}\pmod 3$ so $a_k\equiv a_{k+3}\pmod 3$.

The there are only three equivalence classes each with $3$ elements and so $ a_3, a_6, a_9$ must all contain elements from one equivalence class. There are $3$ choices of which class and $3!$ ways to place the elements. $a_1, a_4, a_7$ must also contain elements from one equivalence class and there are $2$ choices of classes and $3!$ ways to arrange them. and for $a_2, a_5, a_8$ there is one choice of classe and $3!$ ways to arrange them.

So there $3*3!*2*3!*1*3! = 6^4$ ways to do this.

As rotations are considered the same (but not mirror symmetries???) divide by $9$.

So the answer is $\frac {6^4}9$.

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  • $\begingroup$ Rotations are the same, so I think this overcounts by a factor of 3 $\endgroup$ – boink Jul 8 at 5:18
  • $\begingroup$ Since we're placing them in a circle, it's possible we ought to divide by 9 or maybe 18 to get rid of the symmetric options (equivalently, require $a_1=1$, and possibly divide by 2). It's hard to tell, though. $\endgroup$ – Arthur Jul 8 at 5:19
  • $\begingroup$ @fleablood I just meant that the problem statement says they're the same $\endgroup$ – boink Jul 8 at 5:22
  • $\begingroup$ "Since we're placing them in a circle" Placing the them in a circle means that $a_1$ is congruent to $a_8$. It DOESN"T mean that rotations are considered to be the same any more than Mr. Left in a word problem means you need to subtract. But if rotations are the same then divide by $9$. If symmetry is considered the same divide by $18$. $\endgroup$ – fleablood Jul 8 at 5:22
  • $\begingroup$ "I just meant that the problem statement says they're the same" Oh, I didn't see that part. That's one of my pet peeve. If Mr. Left works $8$ hours a day, $5$ days a week how many hours a week does he work. Answer: Well, since the problem contains the word "left" that means we subtract so the answer is $8-5=3$. And question. How many ways are there to place people are a table. Answer: As the problem contains the word "table" and tables are circles rotations are the same.... No, they aren't unless the question says they are. $\endgroup$ – fleablood Jul 8 at 5:28

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