Relaxing at Christmas, it must be time for a puzzle! To play Diffy, draw a square and label each of the corners with some whole number. At the midpoint of each square write the (positive) difference between the numbers at the corresponding vertices. Then draw a new square through the midpoints and repeat the process. Note we ended up with four zeros.
Change the starting numbers, do we always end up with four zeros, and how many steps does this take?
Or perhaps it is rather easier to set this out as follows and use a spreadsheet to explore many possibilities.
I was introduced to Diffy at an excellent lecture by Rob Eastaway and have used this successfully since with students from Year 7 (UK age 11-12) to my Year 13 Further Mathematicians!
Having done a little further research I found an excellent post by Don Steward on Diffy where he has numerous excellent questions for students to explore.
Note The PowerPoint included in Don Steward’s post. We can practise our Algebra too!
Do we need to start with integers?
For further reading and extension, a very thorough analysis, try Diffy Boxes (iterations of the Ducci four number game) by Peter Trapa, September 27, 2006 and Joshua Zucker on Circle of Differences in Numberplay from the New York Times.