0809.1: Tainted Wine

Family Weekend is upon us, and Dean Urgo is preparing a champagne brunch at which 240 bottles of champagne will be served. Unfortunately, some pranksters have tampered with one of the bottles, injecting a magic potion that, though otherwise harmless, will turn the teeth of anyone who drinks even the tiniest drop of it Continental Blue. The person’s teeth will remain dyed for a full week. (The effects of the potion are systemic, and not due to contact between the wine and the teeth.)

The dye is triggered between 8 and 11 hours after drinking, at which time its effects are immediate and obvious. The time of the trigger varies both with the person and with the wine. It is now 6pm on Saturday, and the brunch begins at 10:30am on Sunday.

Dean Urgo wants to find the single bottle that has been contaminated. He is willing to open all 240 bottles of champagne for testing. Since testing only requires the smallest drop, removing any number of drops of champagne will not reduce the quantity in the bottle significantly. The pranksters have also sabotaged the chemistry labs, so the only way to determine if a bottle has been contaminated is by drinking a sip.

Dean Urgo insists on using students to test the champagne. There are over 240 students of drinking age available for testing, but the dean wishes to avoid subjecting more students than necessary to the test. There are 16 students whose parents are not coming to Family Weekend.

Questions:

1. Can Dean Urgo determine which is the single, contaminated bottle using only the 16 students whose parents are not coming to Family Weekend? Provide a solution.

2. Eight of the 16 students whose parents are not coming to Family Weekend have been caught with open containers this term. Can Dean Urgo determine the single contaminated bottle using only these eight students? If so, provide a solution.

3. Can Dean Urgo reduce any further the number of students that must drink from the bottles to be sure to find the contaminated bottle before the brunch begins? If so, provide a solution.

Bonus: Given the minimum number of testers, what is the probability that any one of them will have blue teeth at the beginning of the brunch?