Problem:
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers , then Tadd must say the next four numbers , and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number is reached. What is the th number said by Tadd?
Answer Choices:
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Solution:
Define a round to be the recitations done during the successive turns of Tadd, Todd, and Tucker, in that order. Note that Tadd says number in round , numbers in round , numbers in round , numbers in round , and, in general, numbers in round . In turn, Todd says numbers and Tucker says numbers in round . Therefore numbers are recited by all three children in round . During the course of the first rounds Tadd recites a total of numbers. The sum of this arithmetic series is
During the first rounds, all three children recite a total of numbers. The sum of this arithmetic series is
The number of rounds prior to the round during which Tadd says his th number is the greatest value of such that .
Multiplying by yields . If , then the left side is ; and if , then the left side is . Therefore the being sought is , and Tadd has recited numbers after rounds. During round , Tadd will recite numbers, including his th number.
After 36 rounds have been completed, the children combined will have recited
numbers, the integers from through .
Tadd will say his th number when he has completed reciting numbers in round . This number is .
The problems on this page are the property of the MAA's American Mathematics Competitions