SOLUTION: Given seven equal circles on a hexagonal grid, how many permutations are possible where circles are always adjacent (repetitions such as inversions and reflections excluded)? Th

Question 1176479: Given seven equal circles on a hexagonal grid, how many permutations are possible where circles are always adjacent (repetitions such as inversions and reflections excluded)?
Thank you.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2138)   (Show Source): You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**Understanding the Problem**
We have seven equal circles arranged on a hexagonal grid. The arrangement forms a central circle with six circles surrounding it. We want to find the number of unique permutations of these circles, where adjacency is maintained, and rotations and reflections are considered the same permutation.
**Visualizing the Arrangement**
Imagine the circles arranged like this:
```
O
O O
O O
O O
O
```
Where the center "O" is surrounded by six other "O"s.
**Finding Permutations**
1. **Start with the Central Circle:** The central circle is fixed. We only need to arrange the six surrounding circles.
2. **Permutations of the Outer Circles:** If the outer circles were distinct, there would be 6! (6 factorial) permutations, which is 6 * 5 * 4 * 3 * 2 * 1 = 720.
3. **Accounting for Rotations:** Since rotations are considered the same, we need to divide by the number of rotations. There are 6 rotations (0°, 60°, 120°, 180°, 240°, 300°), so we divide 6! by 6:
* 720 / 6 = 120
4. **Accounting for Reflections:** We also need to account for reflections. In this case, there are 6 lines of reflection symmetry. We already accounted for rotations, so we need to divide by 2 to account for the reflections.
* 120 / 2 = 60
**Therefore, there are 60 unique permutations of the circles.**

Answer by ikleyn(53366)   (Show Source): You can put this solution on YOUR website!
.
Given seven equal circles on a hexagonal grid, how many permutations are possible where circles
are always adjacent (repetitions such as inversions and reflections excluded)?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

        I have another logic/reasoning, and my solution and my answer are different from that by @CPhill.

        But before to start, I'd like to discuss the problem's formulation.

        The problem says

Given seven equal circles on a hexagonal grid, how many permutations are possible where circles are always adjacent (repetitions such as inversions and reflections excluded)?

        I would re-formulate the problem in mathematically more appropriate way:

Given seven equal circles on a hexagonal grid, how many arrangements are possible where circles are always adjacent (arrangements that differ by rotations, inversions, and reflections are considered as indistinguishable).

        Below is my solution for this modified formulation.

S O L U T I O N We consider 7 circles as numbered from 1 to 7 - so the circles are distinguishable. To start, let assume that the circle '7' is in the center. Then, accounting for circular permutations (rotations), we have (6-1)! = 5! = 120 different arrangements. There are 3 different axes of symmetry. To account for reflections, we divide the number of 120 arrangements by 2 three times. We get then 120 : 8 = 15 distinguishable arrangements. We should not make an additional correction for inversion, since the inversion is just accounted as the 180-degree rotation. So, now we only need to multiply 15 by 7 to account for the fact that any of 7 circles can be placed in the center. ANSWER. Under the given conditions (and with my modification) there are 15*7 = 105 different distinguished arrangements.

Solved.

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