Diffy

Introduction

Step 1: Choose any 4 numbers for the top row.  In the example on the right, we have 1, 3, 5, 11.

Step 2: Calculate the positive difference between each pair and write it underneath (including the difference between the first and the last e.g. 11-1 = 10)

Step 3: Repeat this until you get 0, 0, 0, 0.  The row that this occurs is the Diffy Number for that original group, so here we have a Diffy 5 for this particular example


Now repeat the steps above for 4 different numbers, does every set of four numbers end in four zeros eventually?

What is the longest chain you can find?

Instead of using 4 integers (whole numbers), use 4 fractions/decimals/negative numbers

Further Questions and Challenges

For the same set of four starting numbers, how many different “diffy” solutions are there if you change the order of the numbers? e.g. 1, 5, 3, 11 instead of 1, 3, 5, 11

What happens if you add/subtract a constant value to the starting numbers? 

What happens if you double, halve, etc the starting numbers?

What if the starting numbers are based on some rule, e.g. multiples of 3?

What if the starting numbers are all primes?

What about a combined linear transformation?

What about squaring, reciprocals, etc.?

What happens if we start with three or with five numbers?

Are there any patterns if the starting numbers are all odd? All even? Alternating between odd and even?

What if the starting numbers follow linear sequence? A quadratic sequence? A Fibonacci sequence?


Can you work backwards from the last line and find examples of Diffys that reach 0,0,0,0 in:

What do all the Diffy 1’s have in common?

How about the Diffy 2, 3, and 4s?  Why is this?

Why is it hard to find Diffy 6s and above?

If you see a set of 4 numbers, can you predict how many steps it will take to reach 0,0,0,0 without working it out?


For more challenges, take a look at Don Steward's website


Finally, how's this picture below related Diffy?