Question

Interesting and challenging puzzle!
Can you fill the 12 circles with distinct integers from 1 to 12 such that the sum along any line (of the six lines consisting of 4 numbers each) is the same.

Each number is used only once.

Answer 1

There are many possible solutions. One solution is as below.

:) Systematically. .

Take the diagram as 2 triangles: one pointing leftwards and another to the right.
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Derive first that sum of four numbers on each line = 26..
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Start with the leftmost number =1 and the rightmost number = 2.
So sum of ten unknowns = 75.
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Then derive 7 equations for the 10 unknown variables.
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Then derive the sum of two middle (center) numbers to be 23. So they are 11 and 12..
So sum of 8 unknowns = 52.
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Now I have 5 equations for 8 unknowns: as follows:
a+c=14....d+i =13..
e+g=12... h+j = 13..
c+g+h+i = 26..
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Divide numbers 3 to 10 into pairs such that sums of 12, 13, 14 are realized. ..:)
Identify the big ´<´ (less than sign shape ) formed by 2 slanting lines meeting at the leftmost point. The numbers on them are:
1, 5 , 11, 9..
1, 7, 12, 6..
Now identify the two lines forming the big ' >' (greater than sign) meeting at the rightmost point. Numbers on them are:
10, 11, 3, 2..
4, 12, 8, 2..