Just as the topic suggests, I am looking for a quick method of holding numbers for calculations, especially multiplication while using the vertical and crosswise technique, right now I am using my PAO, which while I’ve found it useful for long digit multiplications where speed isn’t my main goal, for short single two by two digits multiplication I find it slow and sometimes exhausting. Is there a better way of holding the numbers?( Especially for the crosswise part, I find it to seem the most unnatural)
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For 2×2 multiplications, I’ve found it’s faster finding a shortcut for each question that minimizes the amount of working memory required.
For example, to solve 73 × 83, I would do apply the difference of two squares formula to get 78² – 5² = 6084 – 25. The 2-digit subtraction requires less working memory than a full cross-multiplication of 73 and 83.
Using these approaches, I can solve arbitrary 2×2 multiplications in an average of 4.0 seconds (with 95% correct) even without seeing the original numbers (e.g. if someone speaks them to me).
For 3×3 or larger where the digits are written, the best method is cross multiplication / criss-cross. That link is to my website, where I explain this method, in case it’s not what you’re already using. Hope that helps!
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Nice idea, I’ve experimented with this difference of squares method before (in fact it was my motivation for learning the 2- digit squares) but I wasn’t fast enough at getting the averages and doing the subsequent subtraction. I will retry this and check how much faster I can get it to be. Thanks
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I think this method I call half-base maybe better though. Basically, if you had to multiply 67×84, you could take 67 to the base 70 and solve in this fashion (84×70)=5880, (3×84)=252 then you do 5880-252 to get 5628. I feel this is easier than the difference of squares method( my major hang-ups with this squares method are the number of operations and the extra steps required for odd sums) but it doesn’t still beat vertically and crosswise method. I just wish I could use vertically and crosswise in my head with enough speed
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I agree that for 67 × 84, because the numbers have different parity (odd/even), you can’t efficiently use the difference of two squares. 75.5² – 8.5² is not easier 
Your method (splitting 67 or 84 into a sum or difference, e.g. 67 = 70 – 3) is far superior in this case.
The pro method is to move a factor of ×3: 67 × 84 = 201 × 28 = 5628 
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In my opinion, the major system is Highly effective.
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