Trachtenberg method multiplication: ×11 and ×12.
Multiplication rules in the Trachtenberg method turn each digit into a small instruction. The ×11 rule is a good first example because you can see the pattern and check the answer easily. The ×12 rule adds one step—doubling the current digit—while keeping the same neighbour idea.
The ×11 rule in one sentence
Start with the rightmost digit. Moving left, add each pair of neighbouring digits. If a sum reaches ten or more, write its final digit and carry 1 into the next step. Finish by adding any remaining carry to the leftmost digit.
Worked example: 497 × 11
Keep the final digit: 7.
Write 6 and carry 1, because 9 + 7 = 16.
Write 4 and carry 1, because 4 + 9 + 1 = 14.
Add the remaining carry to 4, giving 5. Read the answer left to right: 5,467.
497 × 11 = 5,467
Next rule: multiply by 12
Work from right to left. Double the current digit and add the digit immediately to its right. Imagine a zero beyond the right end and another before the left end. Carry tens into the next position, just as in ordinary addition. Unlike ×11, a step can produce a carry of 2.
For 497 × 12, start with 7: double it to get 14, write 4 and carry 1. Next, double 9 and add 7 and the carry: 18 + 7 + 1 = 26, so write 6 and carry 2. Next, double 4 and add 9 and 2: 8 + 9 + 2 = 19, so write 9 and carry 1. Finally, the imagined leading zero contributes 0 + 4 + 1 = 5.
497 × 12 = 5,964
Check it as 497 × (10 + 2) = 4,970 + 994 = 5,964. The neighbour rule is a compact way of organising those place-value contributions.
Multiply by 11, in your head.
Try any two-digit number. The answer appears one small step at a time.
Enter a two-digit number to see the rule.
4 + 3 = 7. Put that sum between the two original digits.
How to practise the rule
Begin with two-digit numbers such as 43 × 11. Say each step aloud: outside digit, neighbouring sum, outside digit. Then introduce examples that create a carry. When ×11 is dependable, try ×12 with both short numbers and carry-heavy ones. The goal is not to rush; it is to make each sequence reliable enough that you notice when something looks wrong.
When you are comfortable, use three- and four-digit numbers. Checking a few answers with standard multiplication helps you learn both the rule and the shape of a sensible result.
What comes after ×12?
Treat each new multiplication rule as a new skill. Learn its wording, work slowly through examples, and only then increase the number of digits. That is the most practical way to learn the wider Trachtenberg multiplication method.
The wider system has other multiplier rules and a general multiplication approach. Its source book is the place to study the full set. For everyday alternatives, compare our multiplication-table tricks.