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# Using only calculations you can do in your head, can you show that the century starting in 3500 is the first that will not contain a year that is a perfect square?

The year 2025 will be a perfect square since 2025=(45)^2. It is the only such year in this century, since (44)^2 = 1936 and (46)^2 =2116. As time goes on, years that are perfect squares will become increasingly more rare.

Using simple calculations that you can do in your head, it is possible to find the first century that does not contain a year that is a perfect square. Doing this requires one simple result from algebra. x^2 – y^2 = (x+y)(x-y). If you get stuck, you can follow the procedure below.

Here are the steps you can use:

1. Show that all centuries before the one starting in 2500 have at least one year that is a perfect square.

2. Show that of the centuries starting in 2500 up to and including the one starting in 3500, exactly one of them does not contain a year that is a perfect square.

3. Show that the century starting in the year 3500 does not contain a year that is a perfect square.

It would then follow that the century starting in 3500 is indeed the first to be squareless.

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