Productive Differences
A slight variant that is particularly good for showing the effectiveness of ‘tracking arithmetic’ is to use the same two by two grid. Add the products along the rows; add the products down the columns; calculate the difference. Your answer in John’s example will be 3.(his first row is 3 4 and second row is 5 6).
Now find four numbers for which the same operation will give you, say, 7. In essentially how many different ways can this be done?
To use ‘tracking arithmetic’, carry out the operations but DON’t Actually DO any arithmetic calculations. Then smile.
Now multiply the two digit number formed by the top row by the two digit number formed by reversing the bottom row. Also multiply the two digit number formed by the first column, and the two digit number formed by the second column reversed. Take their difference. Smile even more broadly!
I don’t know of any generalisation to more digits.
JohnM
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