I too did not bother trying to figure out the logic used in the example displayed in the posts above.I too have been trying to figure out that crazy method for the last three days! LOL! I realized how they filled in the numbers within the squares but could no way figure out how the answer was arrived at along the border. Once you finally get it.....it is pretty neat. However, if you don't know your basic math (add, subtract, multiply, divide) tables how would you know what numbers to place and where. Without those basic skills, this method is useless.
I'd also wager that without the penmanship and neatness skills being taught today, that the average person couldn't draw that grid neat enough to fill in the blanks and have everything lined up to be able to come to the right solution. It would just be simpler and quicker to do it the "tried & true" ol' fashioned way, wouldn't it?
I use pencil/paper for math all the time. Especially when it's quicker than going to find a calculator. It's also nice to have a "paper trail of your calculations" to remember how you arrived at your final answer when you might not come back to that problem for some time. Example: I use a paper trail of the calculations for some of the different herbicides I spray which is especially handy when the same herbicide is used for different applications with different ratios, etc. If a variable has changed it's much easier to see where to make that change in all the calculations while using a calculator. The same applies to all kinds of other uses where math is involved as well.
I think there's a lot of us that still use cash daily. Very few businesses teach their cashiers how to properly count out change......and it's so easy. If a purchase total amounts to $6.79 and the cashier is handed a $20 bill. The $20 bill is laid on the till. The cashier tells the customer the purchase amount ($6.79) then while giving back the change while counting it out: $6.80 (one penny), $6.90, $7.00 (two dimes), $8, $9, $10 (three ones), and $20 (ten $ bill). The cashier does need to know how to count correctly though. And I always watch to see where the cashier puts my payment before he/she enters the sale and starts getting the change. Once in a while they'll make a mistake.
However, anyone that would use a pencil and paper to calculate, using the methods we older guys were taught in school, something like 329.6 x 19.87, rather that just use a calculator, is losing a bit of time that might be better used for something else.
Frankly, even though an old dog, I don't think I could even do it. Division would be even harder for me. Probably impossible For me.
Simpler stuff I just just do in my head. For example, I would round the numbers in my example above, and quickly come up with 6600 as a close enough answer, without use of a calculator, or pencil. (actual number is 6549.152...thanks calculator!)
edit: thinking about it, if the cash register person just put in the amount a person gives him the change should be automatically calculated...unless the cash register is from the same era as many of us! LOL!