AMC 10 Daily Practice Round 2
Complete problem set with solutions and individual problem pages
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 number said by Tadd?
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Define a round as one complete rotation through each of the three children.
We create a table to keep track of what numbers each child says for each round.
| Round Tadd Todd Tucker |
Notice that at the end of the , the last number said is the triangular number.
Tadd says number in round , , numbers in round , numbers in round , and in general numbers in round , At the end of round , the number of numbers Tadd has said so far is , by the arithmetic series sum formula. We therefore want the smallest positive integer such that . The value of will tell us in which round Tadd says his number. Through guess and check (or by actually solving the quadratic inequality), .
Now, using our formula , Tadd says numbers in the first rounds, so we are looking for the number Tadd says in the round.
We found that the last number said at the very end of the round is the triangular number.
For , the triangular number is . Thus the answer is .
