 ## ________________

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?

## Solutions to week 117

In Fifty-Fifty, the total number of balls initially in the bag was k2 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 k2 – 2r + k = 0, giving r = (k2 – k)/2, and b = (k2 + k)/2. This holds for k = 2, 3, 4, 5…The total number of balls is r + b = k2.

Random Toss answer explained:
The probability that the disk lands on one tile only is P1 = (1 – d)2 and the probability it covers four tiles is P4 = πd2/4 so the probability of winning is P = 1 – P1 – P4. 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

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

Come back next week for answers and more puzzles.