In his North Pole Freezer, Santa keeps a hoard of tasty frozen treats to help hi
ID: 3137237 • Letter: I
Question
In his North Pole Freezer, Santa keeps a hoard of tasty frozen treats to help him through the long polar summer. In his stash he puts 45 Fudgesickles, 54 Klondike bars, and 65 Dove bars. Unknown to Santa, some of his elves sneak in from time to time to steal a treat. Each time an elf visits Santa’s cache, he or she takes two different treats but leaves a treat of the third kind. (For example an elf could take one Fudgesickle and one Klondike bar and leave a new Dove bar.) After a few weeks Santa checks on his stash and finds that there is only ONE kind of treat left. What is this treat? What is the largest possible number of this remaining treat and what is the smallest possible number?
Explanation / Answer
Each time an elf visits Santa’s cache, it takes two different treats but leaves a treat of the third kind. Further, there are 45 Fudgesickles, 54 Klondike bars, and 65 Dove bars in Santa’s stash. Since there is only ONE kind of treat left when Santa checks on his stash and since 45< 54< 65, it is apparent that the Fudgesickles would get exhausted. This means that the elves have visited Santa’s stash more than 45 times. Further, during 45 visits of elves, 45 Fudgesickles plus Klondike bars and Dove bars, aggregating to 45 have been removed. Now, since the number of each of the Klondike bars and Dove bars is more than 45,and since there is only ONE kind of treat left when Santa checks on his stash, hence the elves have visited the Santa’s stash at least 54 times. Assuming that during the first 45 visits, the elves remove 1 Fudgesickle and 1Klondike bar each, and in the next 9 visits, the elves remove 1Klondike bar and 1 dove bar each, then the maximum number of Dove bars left in the Santa’s stash would be 65-9 = 56.
However, assuming that during the first 45 visits, the elves remove 1 Fudgesickle and 1 dove bar each, then there would be 54 Klondike bars, and 20 Dove bars left in the Santa’s stash. Further, if in the next 20 visits, elves remove 1Klondike bar and 1 dove bar each, then there would be 34 Klondike bars left in the Santa’s stash.
Thus, the largest possible number of the remaining treat is 56 Dove bars and the smallest possible number of the remaining treat is 34 Klondike bars.