The coach decides to give the team a workout with a couple of probability problems.
From a bag containing red and blue balls, two are removed at random. The chances are 50-50 that they will differ in color.
What were the possible numbers of balls initially in the bag?
Teammate Joachim is tossing circular disks randomly on a floor marked out in squares. He wins if the disk lands so that it covers parts of exactly two or three squares. In the diagram, A and B are winning positions, while C and D are not. Suppose the squares have a unit side and Joachim gets to pick the diameter of the disk.
What probability of winning can Joachim achieve and what disk diameter achieves this?
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Solutions to week 116
Divide by Three
Cutting the Domino
Divide by Three answer explained:
Scale the half hexagon so that its three sides are of unit length. In the first solution, the smaller length is the cube root of 2 and the larger length is the cube root of 4. In the second solution, the three lengths are (9-√33)/12, (√33-3)/6, and (15-√33)/12.
Cutting the Domino answer explained:
In (a1), the pieces are right triangles with legs in the ratio of 2 to 1.
In (a2), the pieces are right triangles with legs in the ratio of 1 to 1.
In (b), the ratio of sizes of the small pieces to the large piece is f = 2-√(3.5) = .12917…
The cut length is C = 5-6√(3.5) = .775028…
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.