Dr. Pepper-
If I understand you correctly:
1) Put a nickel on a flat table.
3) If the six outer nickels are all touching the inner one, there is no way that each outer nickel can touch both adjacent ones, although they will come very close.
4) The total gap between the surrounding nickels is a function of pi. (That space is twice the decimal of PI [multiplied by a constant]).
Is the above correct?
Eventually there will be space(s) large enough for a nickel within the circle, and if the circle is completed by an odd number of coins the director will win.
You can ensure that there is never a circle with an odd number of coins or an unbalanced gap.
By putting your coin exactly opposite his, the gap in the circle must remain divided evenly between the two sides.