Those familiar with Pascal’s Triangle would probably immediately recognize its beginnings in the following:
However, as I was giving this to Katie, I realized that from this much information the pattern is far from obvious. For example, a 4 in the first blank spot seems like a very natural choice to me. Still, I decided to see what Katie would make of this without any ‘hints’. All I told her was that she should find a pattern and continue it.
She pretty quickly filled in the following, staring at the last two blanks and saying that she didn’t know what should go there.
I asked her why she chose this particular way of continuing, but couldn’t quite follow the explanation. I do recall her saying something like “there were 1’s before and so there are 1’s here now.” I said something along the lines of “interesting pattern” and the conversation ended there.
About a week later I stumbled on this piece of paper again and an interesting continuation of Katie’s pattern jumped out at me. Here it is.
In this pattern the triangle contains itself as a sub-triangle. I was pretty excited. I don’t know whether this is what Katie was thinking – probably not quite – but perhaps it was something along these lines. I am now glad that I didn’t give Katie the hint of trying to figure out the blank from the two numbers directly above it. This puzzle turned out to be much more open-ended than I originally thought, and Katie discovered something unexpected. I think that I learned a lesson here :-).