A fun project from Art Benjamin and the Museum of Math

Yesterday Art Benjamin gave a talk at the Museum of Math. One neat tweet from from the talk was this one:

Start with any number whose digits are in increasing order. Multiply it by 9. Add the digits. Did you get 9? #MathMagic pic.twitter.com/gQtPBoBQK7

— National Museum of Mathematics (@MoMath1) October 1, 2016

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:

 

Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.

 

For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.

 

For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.

 

After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

Very neat! Works in any base, too. https://t.co/VBtfRBZgJU

— David Radcliffe (@daveinstpaul) October 2, 2016

It took a bit of thinking for the boys to see what “works in any base” meant, but they did figure it out.

I love this Benjamin’s problem – it makes a great project for kids!