The Trachtenberg Speed System

The general method is the foundation of the Trachtenberg system for multiplication. It works for any two numbers and is based on a "units and tens" criss-cross pattern. You multiply digits in pairs, keeping a running total of the units and carrying the tens.

Example: 34 x 12

• Step 1 (Rightmost digits): Multiply the units digits: 4 x 2 = 8. Write down 8.
• Step 2 (Criss-cross): Multiply the inner digits (4x1=4) and outer digits (3x2=6). Add them: 4 + 6 = 10. Write down 0, carry the 1.
• Step 3 (Leftmost digits): Multiply the tens digits: 3 x 1 = 3. Add the carry-over 1: 3 + 1 = 4. Write down 4.
• Result: Reading from left to right, the answer is 408.

To multiply a number by 12, start from the right. Double each digit and add its "neighbor" (the digit to its immediate right). If there's no neighbor, you add nothing.

Example: 314 x 12

• Digit 4: Double 4 (8). Neighbor is none. Write down 8.
• Digit 1: Double 1 (2) + neighbor 4 = 6. Write down 6.
• Digit 3: Double 3 (6) + neighbor 1 = 7. Write down 7.
• Leading zero: Double 0 (0) + neighbor 3 = 3. Write down 3.
• Result: Reading from left to right, the answer is 3768.

This is one of the easiest rules. Start from the right. The first digit of the answer is the last digit of the number. Then, add each digit to its neighbor on the right. The last digit of the answer is the first digit of the number.

Example: 632 x 11

• Last digit: The last digit is 2. Write down 2.
• Digit 3: 3 + neighbor 2 = 5. Write down 5.
• Digit 6: 6 + neighbor 3 = 9. Write down 9.
• First digit: The first digit is 6. Write down 6.
• Result: Reading from left to right, the answer is 6952.

Start from the right. Subtract the last digit from 10. For the middle digits, subtract the digit from 9 and add its neighbor. For the first digit, subtract 1 from it.

Example: 436 x 9

• Digit 6: 10 - 6 = 4. Write down 4.
• Digit 3: 9 - 3 = 6. Add neighbor 6: 6 + 6 = 12. Write down 2, carry 1.
• Digit 4: 9 - 4 = 5. Add neighbor 3: 5 + 3 = 8. Add carry-over 1: 8 + 1 = 9. Write down 9.
• First digit: First digit is 4. Subtract 1: 4 - 1 = 3. Write down 3.
• Result: Reading from left to right, the answer is 3924.

Subtract the last digit from 10 and double the result. For other digits, subtract from 9, double, and add the neighbor. For the first digit, subtract 2 from it.

Example: 54 x 8

• Digit 4: 10 - 4 = 6. Double it: 6 * 2 = 12. Write down 2, carry 1.
• Digit 5: 9 - 5 = 4. Double it: 4 * 2 = 8. Add neighbor 4: 8 + 4 = 12. Add carry-over 1: 12 + 1 = 13. Write down 3, carry 1.
• First digit: First digit is 5. Subtract 2: 5 - 2 = 3. Add carry-over 1: 3 + 1 = 4. Write down 4.
• Result: Reading from left to right, the answer is 432.

Double each digit. Add half of its neighbor (ignoring remainders). If the original digit is odd, add an extra 5.

Example: 28 x 7

• Digit 8: Double 8 (16). No neighbor. Write down 6, carry 1.
• Digit 2: Double 2 (4). Half of neighbor 8 is 4. Add them: 4 + 4 = 8. Add carry-over 1: 8 + 1 = 9. Write down 9.
• Leading zero: Double 0 (0). Half of neighbor 2 is 1. Add them: 0 + 1 = 1. Write down 1.
• Result: Reading from left to right, the answer is 196.

To each digit, add half of its neighbor. If the original digit is odd, add an extra 5.

Example: 42 x 6

• Digit 2: Digit is 2. No neighbor. It's even. Half of neighbor 0 is 0. Write down 2.
• Digit 4: Digit is 4. Half of neighbor 2 is 1. Add them: 4 + 1 = 5. Write down 5.
• Leading zero: Digit is 0. Half of neighbor 4 is 2. Write down 2.
• Result: Reading from left to right, the answer is 252.

This rule is very simple. The result digit is half of the neighbor. If the original number is odd, add 5.

Example: 468 x 5

• Digit 8: No neighbor. It's even. Half of neighbor 0 is 0. Write down 0.
• Digit 6: Neighbor is 8. Half of 8 is 4. Write down 4.
• Digit 4: Neighbor is 6. Half of 6 is 3. Write down 3.
• Leading zero: Neighbor is 4. Half of 4 is 2. Write down 2.
• Result: Reading from left to right, the answer is 2340.