## Equal Sums on a Circle

Place the numbers 1 to 10 around a circle such that the sum of any two adjacent numbers is equal to the sum of the pair of numbers directly opposite. In how many ways can this be done? Can this be done with 1 to 8? With 1 to 6?

Source: Based on a problem from The Little Giant Encyclopedia of Puzzles.

Solutions were received from Al Zimmermann, Sudipta Das, Dan Ghinea, Dick Saunders Jr., Eugene Bryzgalov, Abhijit Chakankar. There were 24 solutions for N=10, and 2 solutions for N=6. There are no solutions for N=8. Here are some representative solutions:

From Al Zimmermann, for the 24 solutions for N=10:

```1, 4, 5, 8, 9, 2, 3, 6, 7, 10
1, 4, 5, 10, 7, 2, 3, 6, 9, 8
1, 4, 7, 6, 9, 2, 3, 8, 5, 10
1, 4, 7, 10, 5, 2, 3, 8, 9, 6
1, 4, 9, 6, 7, 2, 3, 10, 5, 8
1, 4, 9, 8, 5, 2, 3, 10, 7, 6
1, 6, 3, 8, 9, 2, 5, 4, 7, 10
1, 6, 3, 10, 7, 2, 5, 4, 9, 8
1, 6, 7, 4, 9, 2, 5, 8, 3, 10
1, 6, 9, 4, 7, 2, 5, 10, 3, 8
1, 7, 3, 9, 5, 6, 2, 8, 4, 10
1, 7, 3, 10, 4, 6, 2, 8, 5, 9
1, 7, 4, 8, 5, 6, 2, 9, 3, 10
1, 7, 4, 10, 3, 6, 2, 9, 5, 8
1, 7, 5, 8, 4, 6, 2, 10, 3, 9
1, 7, 5, 9, 3, 6, 2, 10, 4, 8
1, 8, 2, 9, 5, 6, 3, 7, 4, 10
1, 8, 2, 10, 4, 6, 3, 7, 5, 9
1, 8, 3, 6, 9, 2, 7, 4, 5, 10
1, 8, 4, 7, 5, 6, 3, 9, 2, 10
1, 8, 5, 4, 9, 2, 7, 6, 3, 10
1, 8, 5, 7, 4, 6, 3, 10, 2, 9
1, 9, 2, 8, 5, 6, 4, 7, 3, 10
1, 9, 3, 7, 5, 6, 4, 8, 2, 10
```
For N=6:
```1, 4, 5, 2, 3, 6
1, 5, 3, 4, 2, 6
```

Here's a nice logical approach to finding a solution from Abhijit Chakankar:
```Going around clockwise:
1, 10, 2, 8, 4, 6, 5, 7, 3, 9
I could find only one assymetric solution.

Cannot be done with 1 to 8 (in general powers of 4).

Can be done with 6 (powers of 2 that are not powers of 4).
1, 6, 2, 4, 3, 5

The algorithm to get this soln is very simple.

Now perform in loop:
2) pick the next 2 biggest (initially n, n-1) and
put it next to two smallest points (initially next
to 1). Fill their opposite points.
3) pick the next 2 smallest and put it next to the
two biggest points. Fill their opposite points.

For eg. in case of n = 14, you get:

Alg step:  Locations:             Numbers:
1         1, 8                    1, 8

2.1       2, 9, 14, 7             14, 7, 13, 6
3.1       3, 10, 13, 6            2, 9, 3, 10

2.2       4, 11, 12, 5            12, 5, 11, 4

Done.
```

Mail to Ken