(a + b + c)² by substitution, and the square that proves it

0:00You know what to do with the square of a sum of two things. Now somebody hands you three. a plus b plus c, all squared. The honest first thought is that this is a new problem, and that a new problem needs a new proof. It does not, and the reason is worth more than the answer. If you can turn a new problem into an old one in disguise, you do not need a new proof at all. You need a change of name.

0:30Here is the change of name. Write d for b plus c. One letter standing in for two, and nothing else has happened. Because now the expression reads a plus d, all squared. And that is the thing you already know how to do. a squared, plus two a d, plus d squared. Three terms, not six. Nothing new has been assumed. The old identity was applied exactly as it stands, to a and to d.

1:01Now put b plus c back where d was. a squared has no d in it, so it does not move. Two a d becomes two a, times b plus c, which opens out into two a b plus two a c. And d squared becomes b plus c, all squared. Look hard at that last piece, because it is the point of the method. It is the two-letter identity again. The same one.

1:28Nothing new is borrowed to finish. It opens into b squared, plus two b c, plus c squared. Collect everything, and there are six terms. a squared, b squared, c squared. Then two a b, two b c, two c a. Three squares, one for each letter. Three doubled products, one for each pair of letters. Written in that order it sticks. The squares first, then the pairs going round the triangle, a to b to c and back again.

2:02And one thing that trips people up. Two a c and two c a are not two different terms. They are the same term the other way round. There is no seventh term to go looking for. Now the drawing, because the algebra gives the answer and the picture gives the reason. Take a square of side a plus b plus c. Cut it twice across and twice down, at the same places both ways.

2:32Nine pieces. Three of them sit on the diagonal, and those three are squares. a by a, b by b, c by c. The other six are rectangles, and they come in three matching pairs. Two of a by b, two of b by c, two of c by a. That drawing was rebuilt on one thousand seven hundred and twenty eight sets of lengths, and the nine pieces filled the square every time.

3:02Each letter meets itself once, on the diagonal. That is where the three squares come from. Each pair of letters meets twice. Once in this row and that column, once the other way about. That is where the doubling comes from. Which kills a very common guess. There is not one cross term per letter. There is one per pair. With four letters you would get four squares and six doubled products, not four of each.

3:32Two letters give three terms in all, three give six, four give ten, and five give fifteen. Time to make it earn its keep. Square a hundred and nineteen. Split it as a hundred, plus ten, plus nine, and read the six pieces straight off. The three squares: ten thousand, a hundred, eighty one. The three doubled products: two thousand, one thousand eight hundred, and a hundred and eighty. Add them. Fourteen thousand, one hundred and sixty one.

4:06Every three-digit number splits like that. Tried on nine hundred and ninety of them, right every time. Which raises a question about judgement rather than correctness. Seventy eight is eighty minus two. Two parts, three terms, and you are finished. Six thousand and eighty four. A hundred and ninety eight is two hundred minus two. Thirty nine thousand, two hundred and four. Two hundred and fourteen has no round number near it, so it wants the three-way split. Forty five thousand, seven hundred and ninety six.

4:46But a hundred and seventeen can be done either way. A hundred plus ten plus seven gives six pieces to write down. A hundred and twenty, minus three, gives three. Both come to thirteen thousand, six hundred and eighty nine. One of them is half the writing. That is the entire question being asked. Two things about minus signs. The first is that a minus needs no new identity at all. Replace b by minus b, and the same six terms come out with exactly two signs flipped.

5:26The two that flipped are the two containing b. Two a b and two b c go negative. Two c a does not, because b was never in it. The second is a warning. Here is an expression that looks like a three-letter square and is not one. Four x squared, plus nine y squared, plus thirty six z squared, plus twenty four x y, plus thirty six y z, plus twelve z x.

5:56The three squares name two x, three y and six z. The three cross terms want two x, six y and three z. Those disagree, so it is not a square. And here is the sting. Swap the nine and the thirty six, and it becomes two x plus six y plus three z, squared. Swap the twenty four and the twelve instead, and it becomes two x plus three y plus six z. Two repairs, two different answers.

6:30Now a different use for an identity you have had for years. a squared minus b squared is a plus b, times a minus b. Move the b squared across the equals sign and it says something else entirely. a squared is a plus b, times a minus b, plus b squared. Same fact, different job: one factorises a difference, the other computes a square. And you can watch it happen. Take a square of side a. Cut a strip of height b off the bottom, and cut that strip at width a minus b.

7:08Take the long piece, turn it through a right angle, and lay it against the right hand side. It keeps its area - it was a minus b across and b up, and now it is b across and a minus b up. What is left standing is one rectangle, a plus b across and a minus b up, and one small square of side b in the corner, which never moved. Checked on seven hundred and eighty pairs of lengths.

7:39That is worth real money on numbers ending in five. Fifty five squared. Take b to be five. Sixty times fifty, plus twenty five. Three thousand and twenty five, and nothing awkward was ever multiplied. Thirty five squared is forty times thirty, plus twenty five. One thousand two hundred and twenty five. Sixty five gives four thousand two hundred and twenty five. Eighty five gives seven thousand two hundred and twenty five. A hundred and five gives eleven thousand and twenty five.

8:17Now look only at the fronts. Twelve, forty two, seventy two, a hundred and ten. Three fours. Six sevens. Eight nines. Ten elevens. Any number ending in five is ten n plus five, and its square is n times n plus one, in hundreds, followed by twenty five. Tested on four hundred of them. One last thing, and it is a warning about trusting a pattern. Somebody offers you three squares. a plus b minus c, a minus b plus c, and a minus b minus c - and claims they add to twice a squared, plus twice b squared, plus twice c squared.

9:04Test it. Let a and b and c all be one. The left side is one plus one plus one, which is three. The right side is six. It is not an identity. Expand it and the cross terms are still sitting there. They have not cancelled. It comes out right on thirty one triples out of seven hundred and twenty nine. That is not a rule, that is a coincidence.

9:31Now add the fourth one, a plus b plus c squared, and everything changes. Every cross term cancels, in fours. What is left is the square of two a, plus the square of two b, plus the square of two c. Three of those squares do not balance. Four of them do.