## ________________

It’s the season for gift-giving and family visits — and the coach has two puzzles on those themes this week.

## ________________

### Good Relations

Chris says: “Nieces and nephews have I none, but Alex’s father-in-law is my mother-in-law’s son.”

*How are Chris and Alex related?*

### Square Purchases

Last December, the coach purchased several gifts and noted that each one cost a perfect square number of dollars. When the set of prices was written down, every integer from 1 to 9 appeared exactly once.

*If the total cost was the minimum possible, what was the total bill and how many gifts did he buy?*

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## Solutions to week 117

In Fifty-Fifty, the total number of balls initially in the bag was k^{2} for any k = 2, 3, 4, 5… For Random Toss, the disk diameter and the probability of winning are both 4/(π + 4) ≈ .560099153…

**Fifty-Fifty answer explained:**

If there are b blue balls and r red balls in the bag then the probability that two removed from the bag differ in color is 2rb/(r + b)/(r + b – 1) = 1/2. If we define k = b – r then the equation reduces to k^{2} – 2r + k = 0, giving r = (k^{2} – k)/2, and b = (k^{2} + k)/2. This holds for k = 2, 3, 4, 5…The total number of balls is r + b = k^{2}.

**Random Toss answer explained:**

The probability that the disk lands on one tile only is P_{1} = (1 – d)^{2} and the probability it covers four tiles is P_{4} = πd^{2}/4 so the probability of winning is P = 1 – P_{1} – P_{4}. This function is maximized for d = 4/(π + 4) ≈ .560099153… Interestingly, this gives a maximum win probability of P = d = 4/(π + 4) ≈ .560099153…

## Recent Weeks

**Week 117**: Fifty-Fifty & Random Toss, solutions to Cutting the Domino & Divide by Three

**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

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

**Come back next week** for answers and more puzzles.