All about that base – a fun exercise from Art of Problem Solving

Last night I stumbled on a great little exercise from Art of Problem Solving’s Introduction to Number Theory book, though I’d didn’t dawn on me how neat the problem was until later in the evening.

The problem itself is pretty easy to state:  Convert 100 from base 10 to base 9  (or from base 9 to base 8, or base 8 to base 7, and etc.)

What I realized late last night is that working through this problem allows us to find fun ways to connect arithmetic, algebra, and geometry.   So we revisited the exercise this morning,  starting off with a quick review of the original problem:


After the introduction, I wanted to reinforce a basic problem solving idea by starting with an easier problem.  This problem I had in mind was showing how the number 10 in one based could be represented in other bases.  My son suggested looking at the number 1 first, so we did both:


After talking about the easier problems for a little bit we switched from the whiteboard to the floor to see if snap cubes could help us see some geometry.  We reviewed converting the number 10 in one base to one base below to make sure that we understood how the snap cubes represented the different place values.  At the end of the video we took a quick peek at now to represent three digit numbers with the snap cubes.


Now comes the fun!  We know that 100 in base 5 converts to the number 121 in base 4.  Can we see that relationship in a geometric way with our snap cubes?


After seeing the neat geometric connection we returned to the board to talk about the algebraic connection.  This felt like  a really natural way to talk about some basic algebra, and my son seemed to be comfortable with the basic algebra we discussed here:


The next thing we talked through is the question that my son wondered about in the beginning – is there a relationship between these base conversions and  Pascal’s triangle?    We have actually done a few similar projects before.  See here, for example:

Pascal’s Triangle and Powers of 11

I didn’t want to go into too much depth since we were already 30 minutes in to this talk, but we did spend some time looking to see if the next line of Pascal’s triangle – 1 3 3 1 – had a relationship with converting cubes of numbers to different bases:


Having found that Pascal’s triangle did indeed seem to help us understand how to convert cubes into different bases, we went back to the snap cubes to see if there was a geometric connection here, too:


So, a fairly innocent looking question from our Art of Problem Solving book leads to a really fun project connecting arithmetic, geometry, and algebra.  Super fun way to spend a morning.  The more time we spend in this book, the more I’ve come to appreciate how a little introductory number theory can be a neat way to build up number sense.

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