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.
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.