I was practicing some equations using the basic multiplication when I began to wonder if the rule patterns used continued to work on numbers higher than 12. Was expanding the Trachtenberg Basic Multiplication System possible? Why did the system stop at 12?

I started following the pattern used in the system and found it did continue to work, in fact, it was still working when I stopped at multiplying by 34.

So why did Trachtenberg stop at 12? Maybe because children only usually need to learn the multiplication tables up to 12. Also above 12 you will need to more than double a number so the need to know the multiplication tables becomes a requirement to use the method.

## Number Grid

If we arrange the multipliers from 0 to 32 in the following grid we find a couple of interesting features. The first feature is that each row of the grid uses the neighbor in the same way. The second feature is that the members in each column have essentially the same rules.

0 | 1 | 2 | Dont use the neighbor | |||

3 | 4 | 5 | 6 | 7 | use half the neighbor | add 5 if odd |

8 | 9 | 10 | 11 | 12 | use the neighbor | |

13 | 14 | 15 | 16 | 17 | use 1 and a half times the neighbor | add 5 if odd |

18 | 19 | 20 | 21 | 22 | use 2 times the neighbor | |

23 | 24 | 25 | 26 | 27 | use 2 and a half times the neighbor | add 5 if odd |

28 | 29 | 30 | 31 | 32 | use 3 times the neighbor |

I have indicated for each row how they use the neighbor and also "add 5 when odd" is included when using the neigbor includes a half.

## The Extended List of Rules

Here is a list of the rules for the multipliers from 0 to 32 grouped by column from the above table.

0 | don’t use the number or the neighbor |

5 | use “half” the neighbor and add 5 if odd |

10 | use the neighbor |

15 | use one and a “half” of the neighbor and add 5 if odd |

20 | use double the neighbor |

25 | use two and a “half” times the neighbor and add 5 if odd |

30 | use triple the neighbor |

1 | use the number |

6 | add “half” the neighbor and add 5 if the number is odd |

11 | add the neighbor |

16 | add one and a “Half” of the neighbor and add 5 if odd |

21 | add double the neighbor |

26 | add two and a “half” times the neighbor and add 5 if odd |

31 | add triple the neighbor |

2 | double the number |

7 | double the number and add “half” the neighbor and add 5 if odd |

12 | double the number and add the neighbor |

17 | double the number and add one and a “half” times the neighbor and add 5 if odd. |

22 | double the number and add double the neighbor |

27 | double the number and add two and a “half” times the neighbor and add 5 if odd. |

32 | double the number and add triple the neighbor |

3 | Step 1: subtract from 10, double and add 5 if odd Step2: subtract from 9, double, add “half” the neighbor and add 5 if odd. Step3: take “half” the LH digit and reduce by 2 |

8 | Step 1: subtract from 10 and double Step2: subtract from 9, double, add the neighbor Step3: reduce the LH digit by 2 |

13 | Step 1: subtract from 10, double and add 5 if odd Step2: subtract from 9, double, add one and a “half” times the neighbor and add 5 if odd. Step3: take one and a “half” the LH digit and reduce by 2 |

18 | Step 1: subtract from 10 and double Step2: subtract from 9, double, add double the neighbor Step3: double the LH digit and reduce by 2 |

23 | Step 1: subtract from 10, double and add 5 if odd Step2: subtract from 9, double, add two and a “half” times the neighbor and add 5 if odd. Step3: take two and a “half” the LH digit and reduce by 2 |

28 | Step 1: subtract from 10 and double Step2: subtract from 9, double, add triple the neighbor Step3: triple the LH digit and reduce by 2 |

4 | Step 1: subtract from 10 and add 5 if odd Step2: subtract from 9 add “half” the neighbor and add 5 if odd. Step3: take “half” the left-hand digit and reduce by 1 |

9 | Step 1: subtract from 10 Step2: subtract from 9 and add the neighbor Step3: reduce the left-hand digit by 1 |

14 | Step 1: subtract from 10 and add 5 if odd Step2: subtract from 9, add one and a “half” times the neighbor and add 5 if odd. Step3: take one and a “half” the left-hand digit and reduce by 1 |

19 | Step 1: subtract from 10 Step2: subtract from 9, add double the neighbor Step3: double the left-hand digit and reduce by 1 |

24 | Step 1: subtract from 10 and add 5 if odd Step2: subtract from 9, add two and a “half” times the neighbor and add 5 if odd. Step3: take two and a “half” the left-hand digit and reduce by 1 |

29 | Step 1: subtract from 10 Step2: subtract from 9, add triple the neighbor Step3: triple the left-hand digit and reduce by 1 |

## The Key Row

The third row which includes 8, 9, 10, 11 and 12 is the Key Row. This row uses the neighbor.

As you go down a column in the above grid the numbers increment by 5 and for each you an additional half of the neighbor. This gives you a key to quickly work out what rules apply to any of the numbers from 13 to 32.

If you read how I remember the Trachtenberg Basic Multiplication System rules you will understand why I use this as the key row.It is very easy to work with a number that you know the rules for to calculate the rules for the number you want to multiply.

Say we want to multiply by 18. We know that 18 is ten more than 8. Now we know the rules for 8 which are:

- Subtract from 10 and double.
- Subtract from 9 and double and add the neighbor.
- Reduce the left-hand digit by 2

We know that for each increase of 5 we use a half more of the neighbor, and also in this case we use a half more of the left-hand digit in rule 3. As we need to increase by 10 it means we use a whole neighbor more and a whole left-hand digit more.

Now for 18 the rules become:

- Subtract from 10 and double.
- Subtract from 9, double and add double the neighbor.
- Double the left-hand digit and reduce by 2

Not a big change and easy to work out the new rules in your head.

We will try 458 x 18.

**Step 1:** Subtract the 8 from 10 and double.

**Step 2:** Subtract 5 from 9, double and add double the neighbor.

**Step 3:** Subtract 4 from 9, double and add double the neighbor.

**Step 4:** Double the left-hand digit and reduce by 2.

we add the 2 carried from the previous step.

We have our answer of 8244.

We just multiplied by 18 without doing any more multiplication than doubling a number.

I am sure the method would continue to work on numbers higher than 32 but then you are having to do more than triple numbers, which is not necessarily difficult, but it does start to defeat the purpose of not needing to multiply to use this method.

I found this an interesting exercise as it helped me to reinforce the rules for the multipliers up to 12. It also is easier and faster to multiply by some of the higher multipliers this way than using the direct method.

If you have any comments or questions about expanding the Trachtenberg Basic Multiplication System just leave them below.