Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

When playing with my calculator, I noticed the following pattern:

Is there a reason why the last two digits are perfect squares? I know it usually doesn’t work out this way.

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

The answer is: This always happens as long as the tens digits is either 0 or 5.

To see why, let’s expand , where and are nonnegative integers and . If is odd, then the tens digit of will be a 5. But if is even, then the tens digit of will be 0.

Whether is even or odd, we get

The expression inside the parentheses is not important; what is important is that is a multiple of 100. Therefore, the contribution of this term to the last two digits of is zero. We conclude that the last two digits of is just .

Naturally, elementary-school students are typically not ready for this level of abstraction. That’s what I love about this question: this is a completely natural question for a curious grade-school child to ask, but the teacher has to have a significantly deeper understanding of mathematics to understand the answer.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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