Great article from the NY Times on the Danger of Praise, with an excellent puzzle included:

Circle of Differences

The numbers 1, 2, 3 and 4 are written at the corners of a large square.  At each step, at the midpoint of each side, write the positive (or absolute value of the) difference between the numbers at its ends, so in the diagram below, you would write the numbers 1, 1, 1, and 3 at the midpoints of the sides, forming a new (tilted) square.

Joshua Zucker

Then repeat that process, writing the positive difference between the endpoints at the midpoint of each side. Eventually you will reach all zeros.

1. How many steps does it take to get to all zeros?

2. Can you increase the number of steps if you start with different numbers?

Inspired, I created a couple more shapes (triangle and hexagon) in Geometer’s Sketchpad, and quickly made a sheet up for my students, found here.

Do you see any interesting modifications to the questions? What are some solutions?