Roll a Penny Game

You have been given the job of designing the "roll a penny" game for your
school's summer fair.

Background: The player is given a small 'shoot' down which they roll a
coin on to a large board marked with a square grid of lines. The coin
rolls across the board and when it stops, and falls flat, it will lie
either, entirely within a square - not touching a line, or only partly
within a square and across a line. If it is entirely within a square the
player wins. If it is across a line the player loses.

a) Your first effort should provide the player with an equal probability
of winning and losing. If the coin has a radius of one unit, what is the
side length of the square that gives an equal win / lose probability? You
can ignore any element of player skill and any complications associated
with line thickness etc.

b) The school vicar, who knows more about such things than he likes to
admit, points out that the game would be more interesting if the prize
could be made larger. The headmaster, responsible for school finances,
wants the game to at least break even. You are asked to modify your design
such that the player only has a 1 in 10 chance of winning on each roll. If
the coin has a radius of one unit, what is the side length of the square
that gives an 1 in 10 win, 1 in 9 lose, probability? You can still ignore
all complications.

I remember playing the game at primary school in the 1960s and working out
the probability at secondary school, when introduced to the subject, some
fifty odd years ago.

--
David Entwistle