@ascribbler,

You implied that the game can end in a tie when your "solution" gave the probability of a draw as 1/12. The probabilities for the two players must sum to one - one of them has to win. 6/12 + 5/12 != 1.

Here's what you should have written:

The chance they both draw a Square = (6/7)x(5/6) = 30/42 = 5/7

The chance Farmerman draws a Triangle and Leadfoot draws a Square = (1/7)x(5/6) = 5/42

The chance Farmerman draws a Square and Leadfoot draws a Triangle = (6/7)x(1/6) = 6/42

The chance they both draw a Triangle is = (1/7)x(1/6) = 1/42

**The chance that they have to redraw is 5/7 + 1/42 = 31/42**
The total recursive probability of Farmerman drawing a Square and Leadfoot drawing a Triangle is :

1/7 first draw + (31/42)x(1/7)second draw + (31/42)^2x(1/7)third drawer + (31/42)^3x(1/7)fourth drawer + ...

summing the geometric series

(1/7)/[1-(31/42)] = 6/11