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Does 99 x 99 = 9801 or any other (b1) x (b1) = b2  2b + 1 leave you dissatisfied?

Yes. I understand the reason involved.
I am aware that the total discussion is merely exploring a representation system (and about as obvious of one as possible) and not any direct intrinsic property of numbers themselves. I picked the one that allows numbers to be written in the most familiar way of m.s.b. to l.s.b. of traditional reading order. That is why I said that it is not really mathematics.
This is also why I picked this particular subforum to place the thread.
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... because your representation shows all the intermediate steps before the final summation, whereas the product shows only what's left after the summation.

Yes. Essentially I just tried to be a little more compact than plopping down N grains of rice to represent N. No calculation is done. No adding. No carries. I merely organized it such that a product can be written with one factor along each of its dimensional borders. That allows reading the factors on the edges in a glance. Of course the factors must be known to draw it that way. But it suggests drawing a border around a mess of 1s and 0s and asking oneself if any manipulation of the interior can move the bits such that it can be turned into what I called the canonical form (with factors along the borders and the entire content a valid Boolean "And" table as is intrinsic to the implied binary long multiplication). This is intended to develop a feel about numbers themselves. There are obvious mathematical rules about how to move the 1s around such as that they double or combine when moving to adjacent magnitude positions.