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Suppose I have a 3x3 grid and the integers 1 to 9 inclusive. I want to find out how many ways are there to arrange the numbers.

My question is: Do I need to account for double counting?

| 1 | 2 | 3 |

| 4 | 5 | 6 |

| 7 | 8 | 9 |

| 3 | 6 | 9 |

| 2 | 5 | 8 |

| 1 | 4 | 7 |

where if I rotate the grid such that the row becomes the column and the column becomes the row.

I understand the numbers are distinct, but the configuration is essentially the same?

If so, then is total no. of ways of arranging = 9! / 4 = 90720 ? (because I can rotate the grid 4 times)

Lau Di Qing
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  • You need to define which configurations are equivalent, then count the inequivalent ones under whatever rules you select. You can certainly specify that you want to count those which are related by rotation as equivalent and dividing by $4$ is correct in this case. You can also say you consider reflections equivalent and (if this is including rotations) you should divide by $8$. Maybe you also consider changing $n$ to $10-n$ as equivalent. This is more complicated because there are some diagrams that get transformed to an equivalent one, so a simple division does not work. – Ross Millikan Aug 21 '22 at 07:14
  • A lot depends on the context. Did you create this question, or was it on an assignment? If you created the question, then it's up to you if you want to count arrangements only up to rotation. If it was on an assignment, then you need to either look more closely at the question (if there's more information there that you didn't write above), or ask the creator or instructor for clarification. – Brian Tung Aug 21 '22 at 07:26
  • @RossMillikan Thank you for your insight, I realised I have not taken into consideration reflections – Lau Di Qing Aug 21 '22 at 08:30
  • @BrianTung Thanks, I will seek clarification on this "ambiguity" – Lau Di Qing Aug 21 '22 at 08:32

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