Arrange a bunch of, say, coins in a square grid. If there are *n* rows and *n* columns of coins, there will be *n*^2 coins in total. Now remove one row of *n* coins and use these to add an extra column. But, because you now have one fewer row, you only need *n*−1 coins for the new column. You have one coin left over. This demonstration works for any whole number *n* from 2 up.

Just to pile on, multiplication is just compact notation for addition. 5*5 is a compact notation of 5+5+5+5+5. Now imagine it’s a big number and you don’t have a calculator, if you remove a 5 what happens? It goes down by 5. What happens if you change all the remaining 5s into 6s? It goes up by the number of numbers. Put it together: if you take a square, lower one factor and raise the new opposite the result is a subtraction of the original factor and addition of the exactly once lowered factor. So 5^2 minus 5 plus (5-1) is 24 or (5^2 )-1.

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