Mental multiplication - via the Trachtenberg cross-multiplication method

Note This technique is much easier once you have become fluent in the Major-system memory images, so you are advised to practice that module first before attempting this.

In this technique we get single-digit temporary carries. It is suggested that you remember these using the number-shape system, so for example swan for 2 and ship for 7. The calculation results in double-digit permanent values that form part of the answer, so it is suggested that you remember these using the Major system, so for example 'curry' for 74.

The basic procedure is shown in the following example:

3 4 ×
6 3

1. Multiply 3 by 4 to give 12. This is memory image 'swan' for the 2, with a carry.
   
2. Cross-multiply 3 x 3 to give 9 and 4 x 6 to give 24. Add them: 33.
   
3. The right-hand '3' of the 33 is added to the carry making 4. Join this to the 2 to make 42. This is memory image 'rhino.'
   Remember this, as this part of the calculation is now done, and the image does not change.
   However you now have the left-hand 3 of the 33, so remember this temporarily with 'camel.'
   
4. Multiply the 3 and the 6 to give 18. Add 18 to our 'camel' carry, which was 3, to give 21. 21 is a 'net' so we now have:
   net rhino which is 2142 , which is the answer.
   
   

For longer-digit multiplications, say for triple-digits, you have to do a triple-multiplication for the middle value. Proceed ass per the following example:

2 7 3 ×
4 2 5

1. Multiply 3 by 5 to give 15. This is memory image 'hook' for the 5, with a carry.
   
2. Cross-multiply 3 x 2 to give 6 and 7 x 5 to give 35. Add them: 41.
   
3. The right-hand '1' of the 41 is added to the carry making 2. Join this to the 5 to make 25 This is memory image 'nail.'
   Remember this, as this part of the calculation is now done, and the image does not change.
   However you now have the left-hand 4 of the 41, so remember this temporarily with 'yacht.'
   
4. Now the 3-way cross-multiplication: cross-multiply the top-left 2 with the bottom 5 to give 10.
   multiply the top 7 with the 2 directly underneath to give 14. Add this to the 10 to give 24.
   multiply the top left 3 with the bottom 4 to give 12. Add this to 25 to give: 36.
   
5. The right-hand 6 is added to the previous carry which was yacht,i.e. 4, giving 10, which means 0 with carry. The carry is added to the 3 to give 4. 40, which is a 'rose,' is our temporary 'middle' section.. So in your head you should now have 'rose nail.'
6. Cross-multiply the top 2 with the bottom middle 2 to give 4, then the 7 with the 4 to give 28. Add together gives 32.
7. The 2 of the 32 (moon) is added to the 4 of the rose, to give 6. The middle section, 60, is now done. 60 is 'cheese,' but remember the 3 of 32 as 'camel.'
8. Lastly multiply the right-most 2 by 4 to give 8, and add it to our camel, 3, giving 11. This is a 'Teddy.'
9. Our final answer is teddy, cheese, nail , which is 116025.

Difficulty Levels - typical questions:

1: 12 x 18
2: 33 x 15
3: 362 x 578