Sabine gives part of her collection of shells, perfect square in position
Source: Dutch NMO 2020 p5
November 23, 2020
combinatoricsnumber theoryPerfect Squares
Problem Statement
Sabine has a very large collection of shells. She decides to give part of her collection to her sister.
On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day.
The th day is the first day that she ends up with fewer than shells. The th day she ends up with a number of shells that is a perfect square for the tenth time.
What are the possible numbers of shells that Sabine could have had in the very beginning?