At a local country fair, a volunteer fire department has a fundraising booth with a game of chance. In the center of the booth there is a large square board with 100 non-overlapping red circles of equal size. Players throw nickels (5 cent coins) and the object is to have your coin come to rest completely inside a circle.

While playing the game I wondered what would happen if one were allowed to throw dimes (10 cent coins) instead. Being smaller makes it easier to land inside the red circle, but being of higher denomination you are throwing more money away^{*}.

Is there a size circle for which it wouldn't matter?

A nickel has a 21.21mm diameter whereas a dime is only 17.91mm. The board looked to be a square about 8 feet on a side. Any other assumptions may need to be left up to you.

^{*}Quite literally, but the money is for a good cause. The prize is a box of candy worth about a dollar.

Consider two cases: 1.
5c coin, 2. 10c coin.

Let us ignore all data that is equal for both cases:
quantity of circles, booth dimension, the nominal value of the prize, the exact value of pi
etc.

The relevant circle to a painted **goal circle** is one
of diameter D decreased by d of the coin.

So the probability for a coin of diameter d is proportional
to (D-d)^2.

The value of coin in case 1 being .5 of the value of coin in case 2, we get:

(D-d2)/(D-d1)=sqrt (2).

So D=(d1*sqrt(2)-d2)/( sqrt
(2)-1)

Inserting the numerical values one gets

**D(eq)= 29.177 mm**

answer.

**Looks too small to attract customers.**

If I was the booth owner I would prefer circles of a 50mm diameter distributed over an area 30 times bigger than the combined area of all 100(or another number) existing circles.