I'm not quite sure if this is the right place to ask this question, but I figure someone might have heard about the following, which I was told as a motivating story in an undergraduate intro to algorithms class years back. Unfortunately, I have no means of obtaining a source on this account:
In Roman times, when going to the market, it would be a commonplace problem to multiply e.g. a price with a quantity of an item to calculate the total price. Since people weren't able to do grade-school multiplication, let alone in their heads, this posed a serious problem. One solution would have been to produce boards that showed a multiplication table. But the quadratic space (weight, production time, etc.) requirements made this not viable.
However, Romans could do addition, subtraction (using some aids, possibly), and division by two, relatively easily; at least compared to outright multiplication of arbitrary numbers. So the clever solution the Romans came up with was to produce boards (of linear size in the size of the range of numbers to be captured) that contained a list of squares. Then, they could employ $ab = ((a+b)^2 - a^2 - b^2)/2$ to perform multiplication, reading off the squares from their table and adding/subtracting as they knew how to do it.
Is this something the instructor made up, or has anyone ever heard of this practice? If so, I would be extremely happy about a source for this.
The same story exists with the Romans replaced by the Babylonians. There is not a single trace for either claim.
Both Babylonians and Romans used an abacus for doing calculations; in the case of the Romans, such instruments were found, for Babylonians investigations of calculation errors at least suggest the use of a primitive abacus.