Question 1135387
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Moving 2 coins can make a circle; moving only 1 can't.<br>
Think of the 6 coins when they are in a circle as the vertices of a regular hexagon.<br>
The problem then becomes determining the minimum number of coins that have to be moved from the original triangular array to accomplish that.<br>
Then the problem is to find the largest number of the 6 coins in the original array that do NOT need to be moved to become vertices of a regular hexagon.<br>
A look at the original array shows that 4 of the 6 coins can stay where they are.  For example....<br><pre>

        1                    6   1

      2   3        -->     2        3

    4   5   6                4   5<br></pre>