Math Investigator 3.3

NUMBER CHAINS on the Math Investigator prints a number chain by multiplying the units digit of a chosen number by any whole number less that 50 and adding the product to the number formed by the remaining digits.

The following task was given to an elementary school class for practice in multiplication. Start with a two-digit whole number, double its units digit, and add its tens digit to obtain a new number. Repeat this process with each new number. For example, the number chain shown below starts with 15, and the second number in the chain is 2 x 5 + 1 = 11. This number chain was continued until the first number in the chain was obtained. A number chain is complete when any previous number in the chain is repeated.

15 -> 11 -> 3 -> 6 -> 12 -> 5 -> 10 -> 1 -> 2 -> 4 -> 8 -> 16 -> 13 -> 7 -> 14 -> 9 -> 18 -> 17 -> 15

1. The number chain in the above example has all the whole numbers from 1 to 18. What happens if we begin the number chain with 19? With a number greater than 19?
2. Suppose that instead of doubling the units digit we use a multiplier of 3 (or any greater number) and add the product to the number formed by the remaining digits. Will a number chain be produced?
3. Find a few patterns and write some conjectures about number chains.
4. Using a multiplier of 2 or 3 produces just one chain to which all numbers go. However, a multiplier of 4 produces several chains. Investigate the multipliers 4, 5, 6, 7, 8, and 9. Which of these multipliers produce just one number chain and which produce several number chains? For these multipliers find the numbers that have just one number in their chain (go to themselves in one step). Form a conjecture about such numbers.
5. Some interesting results occur when the multiplier is 10. Investigate. What types of chains are produced and what numbers go to themselves in one step when the multiplier is 10? Explain the reasons for your results.
6. Investigate chains for multipliers greater than 10. Look for patterns and form conjectures for predicting the outcomes.