Varsity Math, Week 117

Varsity Math 117

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The coach decides to give the team a workout with a couple of probability problems.

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Disc

Fifty-Fifty

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?

Random Toss

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?

Random Toss Squares

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Solutions to week 116

Solutions for Divide by Three and Cutting the Domino are shown below.

Divide by Three
Divide by Three

Cutting the Domino
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.

Divide by Three

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…

Cutting the Domino

Recent Weeks

Week 116: Cutting the Domino & Divide by Three, solutions to Thanksgiving Split & Easy as Pie

Week 115: Thanksgiving Split & Easy as Pie, solutions to Primes and Products & Two Pints of Cider

Week 114: Primes and Products & Two Pints of Cider, solutions to Prime Presents 1 & Prime Presents 2

Week 113: Prime Presents 1 & Prime Presents 2, solutions to Precise Prescription & Gift Dilemma

Week 112: Precise Prescription & Gift Dilemma, solutions to Pumpkin Crop & The Halloween Goblin

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.