For an introduction to this series see this article. Briefly, I discuss some problems in mathematics that can be approached and even answered with only elementary mathematics.
Today’s post was inspired by a question from James Tanton (@JamesTanton on Twitter) who frequently asks interesting math questions. The question
Find all perfect squares whose base ten representations are a single repeating digit.
Now, before going on, I very strongly urge you to try to solve this problem. I will give you a little hint: All you need to solve this are the basic arithmetic operations that you learned in the first years of elementary school (plus some ingenuity).
Go play. The article will be here when you get back.
Well? Did you figure it out? Did you come up with anything interesting? Maybe your solution is different from mine. Tell me in the comments. In the meantime, here is what I did. I played. At first, I didn’t see how to approach this problem. So, I wrote a program to look at the squares of the 1st 200 numbers. Except for 0 and 1, none of them were a single digit. Hmmm. I could have looked at higher numbers, but that would not have been proof.
Then I thought about what the final digit of a square number could be. Because the last (or units) digit of a product (including a square) is only affected by the units digit of the product, there are only certain numbers that a square can end in. That is:
- If a number ends in 0, its square ends in 0
- If a number ends in 1 or 9, its square ends in 1
- If a number ends in 2 or 8, its square ends in 4
- If a number ends in 3 or 7, its square ends in 9
- If a number ends in 4 or 6, its square ends in 6
- If a number ends in 5, its square ends in 5
Now, if you didn’t figure out an answer in the first intermission, I urge you to take a second one. I’ve given you another hint.
After this, I fumbled around for a while. But then I had a thought: For a number to be all one digit, its last 2 digits have to match. And the last two digits of a square can only be affected by the last two digits of the number that is squared. Then I looked at the squares of 1 to 100: None end in consecutive digits other than 0; but a square number can’t be all 0’s unless the number squared is 0.
So, no number squared is all one digit, repeated, except for the single digit squares (0, 1, 4, 9) which are kind of beside the point.