Sum 20
How many ways are there to place a distinct integer from 1 to 9 into each red circle such that the sums of four red circles on each blue line or green circle are all equal to 20?
Clarification: The rotations and reflections that can be obtained in the diagram are considered the same solutions.
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1 solution
3 green circles of 20 each sum to 60.
3 blue lines of 20 each sum to 60.
The red (orange?) circles around the outside are counted twice each.
The red circles on the inside (the vertices of the triangle formed by the blue lines) are counted four times.
The sum of the numbers from 1-9 is 45.
If we call the inside numbers x , y , z :
We have:
2 ∗ 4 5 + 2 ( x + y + z ) = 1 2 0
x + y + z = 1 5
From here I started experimenting with different options and found the 4 solutions.