The coach poses a medical challenge to the team: A doctor must measure out a precise fraction of a tablet for a young patient. She has an empty 5-ounce vial, an empty 3-ounce vial, a water supply, a sink, and a single water-soluble tablet.
How can she measure out a dose (dissolved in water) of exactly 10% of a tablet?
Math team members Mandy and George want to buy a DVD set for the coach, and each has a positive whole number of dollars. Mandy finds that she is $2 shy of the price; George finds that he is $32 shy. If they pool their money, they still don’t have enough to buy the DVD set.
How much is the DVD set if it costs a whole number of dollars?
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Solutions to week 111
Pumpkin Crop answer explained: Let the weight of the total crop be W. The 42 lightest pumpkins weigh an average of .25W/42 each and the 50 heaviest pumpkins weigh an average of .30W/50 each. Thus, the remaining n pumpkins weigh .45W and the average weight of each of those pumpkins must be .45W/n. This must not be less than .25W/42 nor more than 30W/50. This leads uniquely to n = 75 so the total number of pumpkins in the crop was 75 + 42 + 50 = 167.
The Halloween Goblin answer explained: What is needed is agreement from all that whoever goes first will be a martyr. All of you agree on an ordering of cards that labels the full deck from 0 to 51. The martyr adds the numbers of the cards he or she can see and takes the result modulo 52 (divide the sum by 52 and take the remainder). Call this result R. The martyr guesses the card corresponding to R. Each other person then knows that the sum of cards they see, leaving out the card from the martyr, will equal R if their card is added in. Each person other than the martyr deduces their card and is set free.
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.