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In the figure, place the numbers 1-12 at the locations a-l such that
each side has the same sum.
There are possibly many solutions.
Can you find a solution for each possible common sum?
To aid in comparisons, when submitting answers, minimize 'a', then 'b'. |
a b c l d k e j f i h g |
Here's a complete list of solutions, with rotations and reflections discarded.
Sum = 17
1 7 9 1 9 7 1 10 6
11 6 11 4 11 4
5 2 5 6 5 7
8 12 10 8 9 8
4 10 3 2 12 3 3 12 2
Sum = 18
1 7 10
9 3
8 5
6 11
4 12 2
Sum = 19
1 6 12 1 7 11 1 10 8
10 2 10 5 12 4
8 5 8 3 6 7
4 11 2 12 2 9
7 9 3 9 6 4 11 5 3
1 10 8 2 8 9 3 9 7
11 6 10 6 12 1
7 5 7 4 4 11
9 2 1 12 10 2
3 4 12 11 5 3 5 8 6
Sum = 20
1 7 12 1 9 10 2 6 12
11 3 11 4 8 3
8 5 8 6 10 5
2 9 7 2 1 11
10 4 6 5 3 12 9 7 4
2 8 10 2 8 10 2 11 7
12 1 11 4 12 5
6 9 7 6 6 8
3 7 1 9 4 3
11 5 4 12 3 5 10 1 9
Sum = 21
3 6 12
10 4
8 5
2 7
11 1 9
Sum = 22
4 6 12 6 4 12 6 5 11
7 2 9 2 9 1
11 8 7 8 7 10
1 5 5 3 3 4
10 3 9 10 1 11 12 2 8
The solutions for sums 20, 21 and 22 can be derived directly from the
solutions for sums 19, 18 and 17 (respectively) by replacing each number N in
a solution with 13-N. So, in a sense, there are only half as many distinct
solutions as I show above.