# 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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## 13 Answers

For the algebra, it may be simpler to use the fact that (x+1)^2 – x^2 = 2x+1

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3600 = 60 squared. 59 squared = 60 squared – 60–59 = 60 squared minus 119.

The difference between the square of a number and its next higher square is actually twice that number +1.

60^2 -> 3600

59^2 = 3600 – 119 = 3481

58^2 = 3491 – 117 = 3364

57^2 = 3364 – 115 = 3249

56^2 = 3249 – 113 = 3136

55^2 = 3136 – 111 = 3025

54^2 = 3025 – 109 = 2916

53^2 = 2916 – 107 = 2809

52^2 = 2809 – 105 = 2704

51^2 = 2704 – 103 = 2601

50^2 = 2601 – 101 = 2500

49^2 = 2500 – 99 = 2401

note that from there on down the difference between the current and the next lower square is less than 100. So it will be impossible to skip a century with going down one square.

Since every century between 2500 and 3500 has a s square int them and the period from 3500 through 3599 doesn’t, this is the first period to not have square in them.

@whitenoise , You got most of what I was looking for. You can also avoid computing the squares of the numbers greater than 2500, which I will show later.

This solved your question without having to think much about it… call it laziness. :-)

I look forward to your explanation.

A century actually starts on xx01, so the first century to not have a perfect square would be the 2500’s.

60*60=3600

59*59=3481

@amujinx correct,

50*50=2500

51*51=2601

What’s it like to be good with math AND big words?

Guys…playing authistic?

You knew perfectly well what the question was meant to ask…

I would mentally compute squares as

x^2 = (x-y)(x+y) + y^2

The value of y can be picked strategically. I pick y = x – 50.

61^2 = 50*72 + 121 = 3721

60^2 = 50*70 + 100 = 3600

59^2 = 50*68 + 81 = 3481

58^2 = 50*66 + 64 = 3364

Here, you can see that a century is skipped whenever the hundreds digit decreases by at least two. The first term of the formula (the 50 times N) always decreases in value by one hundred. So, a century will be skipped whenever the second term (the small square) decreases its hundreds digit.

Below 59^2, this is not possible, as the second term is already less than 100. So, the jump between 3481 and 3600 is the smallest place where an entire century is skipped.

The 36th Century would begin with 3501, so 3600 would qualify, but since you phrase the question this way, the easy trick is:

60×60 = 3600.

Subtract 60 + 59 from 3600, and you have a number under 3500, 3481 to be exact.

Yes, I studied accounting in school

@filmfann , Precisely what I had in mind for the third step.

The second step is just a matter of counting First realize that starting with 2500, every century has at most one perfect square, Stated differently, every century has 1 or 0 perfect squares, 2500 to 3599 is 11 centuries. The perfect squares in this period are the 10 squares of the numbers from 50 to 59. That means 10 of the centuries have 1 perfect square and 1 century does not have any. Then using @filmfann ‘s approach, you can show that the last century in the group is the one missing the perfect square.

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