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

 What’s the difference between a two-digit number and its reverse?

Choose a two-digit number, e.g., 20. Make a new number by reversing the digits, e.g., 02. Subtract the smaller number from the larger number, e.g., 20 - 02 = 18. Check your subtraction by adding your result, e.g., 18 + 02 = 20. Do you notice a pattern? If not, choose another two-digit number and go back to step 1.

On the basis of your findings, see if you can predict the difference between a two-digit number and its reverse.

For more of a challenge, investigate three-digit numbers.