Here's a great question from an essay by mathematician Paul Lockheart called "Mathematician's Lament" https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
One interesting way to think about numbers is as piles of stones. So, instead of thinking of the number 3, you can imagine three stones in your mind, or draw three dots on a piece of paper.
Thinking about numbers in this way can be useful because of the different ways we can arrange the stones. A 3 can be arranged with all stones in a row
..., or as a triangle
:.. By considering these arrangements, we have a different way of thinking about the number.
So, for instance -- if you take an even number, you'll be able to arrange the stones into two equal rows. For an odd number, one of the rows will have a stone that doesn't have a match:
6 ::: 9 ::::.
Keeping this in mind, consider the following sequence:
1 1+3=4 1+3+5=9 1+3+5+7=16 1+3+5+7+...
1, 4, 9, 16 ... these are some very particular numbers! In fact, they are the first four squares:
1=1x1 4=2x2 9=3x3 16=4x4
So, is it true that this sequence will always produce a square number? And why would that be the case?