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, in the students’ original handwriting:
Here’s the explanation that my students told me (but didn’t write down): they wanted me to add adjacent numbers on the bottom row to produce the number on the next row, building upward until I reached the apex of the triangle. For example, the lower-left portion of the triangle would build like this (since 1+4=5, 4+9=13, 9+16=25, etc.):
5 13 25
1 4 9 16
Then, after I reached the top number, they wanted me to take the square root of that number. (Originally, they wanted me to first multiply by 80 before taking the square root, but evidently they decided to take it easy on me.)
And, just to see if I could do it, they wanted me to do all of this without using a calculator. But they were nice and allowed me to use pencil and paper.
And I produced the answer in less than five minutes.
I’ll reveal how I got the answer so quickly in this series. In the meantime, I’ll leave a thought bubble if you’d like to think about it on your own.