Math Delights

Adapted from http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=148

Create caterpillar numbers using the following procedure:

Pick a number, e.g., 11. If your number is even, then divide the number in half. If your number is odd, then add 1. If the last number is greater than 1, go to step 2.

For example, starting with 11, produces the caterpillar 11 - 12 - 6 - 3 - 4 - 2 - 1.

Investigate caterpillar numbers. Consider the following questions:

What's the shortest caterpillar? What's the longest caterpillar you have found? What caterpillar has the pattern odd-even-odd-even-odd...even-odd...? Can you create caterpillars from tail to head? What's the smallest head (1st number) of a caterpillar of length 6? What's the largest head of a caterpillar of length 6? Are there any caterpillar numbers that never end?

Source: J.M. (Bellingham), who led me to J. Russell, "Caterpillar Collection," Mathematics Teaching, March 2005

nrich.maths.org/991

Materials: sugar cubes, felt pens, and worksheets

Take 8 sugar cubes and arrange them into a 2x2 cube. Paint all 6 of the 2x2 sides. How many of the resulting of the resulting unit cubes are completely unpainted? How many are painted on just one side? How many are painted on two sides? How many are painted on three sides? There are 8 sugar cubes. Each face of each cube is to be painted either green or yellow. Work out how to paint the faces so that the cubes can be put together to make a 2 x 2 cube that is Green all over AND can be rearranged to make a 2 x 2 cube that is Yellow all over.

Note: The sugar cube activity was popular perhaps because the children liked the idea of playing with something sweet. We let them eat one sugar cube each at the end of the activity.

Note: The following book contains many more activities involving cubes.

Mathematics with Cubes: Problem Solving Activities for Older Children www.amazon.com/Mathematics-Cubes-Problem-Activities-Children/dp/1871098149