[Preface NOTE: The material presented below was lifted from an article I saw some time ago by Dan Brutlag, “Making Your Own Rules”. THE MATHEMATICS TEACHER, November 1990. pp. 608-611. What is being given below is how I prepared a handout to give to my classes, mostly as anactivity for use by a substitute.]

#### NUMBER SEQUENCES

Students: I found the information below in a math magazine. I think you will find it interesting, as I did.

YOUR ASSIGNMENT:

- Study/investigate the results of applying the rules below to many different numbers. Keep good records to see if something “
*special*” always happens. Describe what you find.
- Then make up a set of rules of your own to investigate, similar to or completely different from the samples given. Use the table of ideas that you find at the end of this handout to help you with this. Write up a little report about what you discover.

A. Lori’s Rule

- Start with any three-digit number.
- To get the hundreds digit of the next number in the sequence, take the starting number’s hundreds digit and double it. If the double is more than 9, then add the double’s digits together to get a one-digit number.
- Do the same thing to the tens and units digits of the starting number to obtain the tens and units digits of the new number.
- Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 567, 135, 261, 432, ???, ???, …

B. Barbette’s Rule

- Start with any three-digit number.
- Add all the digits together and multiply by 2 to get the hundreds

and tens digits of the next number in the sequence.
- To get the units digit of the next number, take the starting

number’s tens and units digits and add them. If the sum is more than 9, add the digits of that sum to get a one-digit number for the units place.
- Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 563, 289, 388, 387, ???, ???, …

C. Ramona’s Rule

- Start with any three-digit number.
- Obtain the next number in the sequence by moving the hundreds digit of the starting number ito the tens place of the next number, the tens digit of the original number into the units place of the next number, and the units digit into the hundreds place.
- Then add 2 to the units digit of the new number; however, if the sum is greater than 9, use only the units digit of the sum
- Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 324, 434, 445, 546, ???, ???, …

D. Lisa’s Rule

- Start with any three-digit number.
- If the number is a multiple of 3, the divide it by 3 to get a new number.
- If it is not a multiple of 3 then get a new number by squaring the sum of the digits of the number.
- Repeat Steps 2 and 3 as often as necessary to find the special happening.Example: 315, 105, 35, 64, ???, ???, …
Example: 723, 241, 49, 169, 256, ???, ???, …,

** **

**TABLE OF ATTRIBUTE LISTS**

**NUMBER OPERATORS** |
**NUMBER PROPERTIES** |

Add ___ to ___ |
Divisible by ___ |

Move to another position |
Hundreds, tens, or units place |

Take positive difference |
Prime |

Double |
Composite |

Square |
Even, Odd |

Halve (if even) |
Multiple of ___ |

Multiply by ___ |
Greater than |

Replace by ___ |
Less than |

Exchange |
Equal to |

###### Postscript (7/29/99)

Here is a personal sidelight to this activity. On April 23, 1991, I designed my own sequence rule-set. Here it is.

**TROTTER’S RULE**

- Start with a four-digit number.
- To get the thousand’s digit and hundred’s digit of the next number, double the sum of the first three digits of the original number, that is, the thousand’s, hundred’s, and ten’s digits. (If this result is only a one-digit number, append a 0 to the left of that number, so a “6” would become “06”.)
- To get the ten’s and unit’s digit of the next number, double the sum of the last three digits of the original number, that is, the hundred’s, ten’s, and unit’s digits. (If this result is only a one-digit number, do as was done in Step #2.)