Two Plus Two Equals . . . Two?

Once Brian had figured out how to draw the anti-tic-tac-toe game whether he went first or second, he and Ozwald (the fellow from BoB) fell to talking of other things. Brian was starting to wonder about his friends, though. Their lovely feast was supposed to come out in another quarter of an hour, and there was still no sign of them.

Had he but known it, they were making their battered and bloody way to the inn, having gotten free of their chains just a few minutes before. Because we are omniscient we know this, and aren't worried about their lovely dinner. Pity the poor protagonist.

That is all by the way, though.

Ozwald observed Brian's impatient distraction. Having decided he liked Brian (a conclusion he reached around the time that Brian offered him a drink), he was determined to distract the younger man from whatever was bothering him.

"How about a chance to win a few of your coppers back?" suggested Ozwald.

With an effort, Brian dragged his mind back to the table, away from absent friends and the lovely roast that he had ordered for them.

"Sure . . . !" He knew enthusiasm was called for, but it was hard to muster.

"Well, I have a little mathematical problem here. You take a look and figure out what's wrong with it. If you figure it out before the clock strikes the hour in another few minutes, I will give you half your coppers back. Otherwise, you pay me half as many. See, it's even in your favor!"

Brian somewhat uncertainly took the grubby bit of paper that Ozwald proferred. With one last glance at the door he bent his attention to the scrap. And saw:

a = b + c
a (a - b) = (b + c) (a - b)
a^2 - ab = ab + ac - b^2 - bc
a^2 - ab - ac = ab - b^2 - bc
a (a - b - c) = b (a - b - c)
a = b

Brian frowned over it.

"The math looks ok . . . " he said somewhat doubtfully. The math did look ok, but the conclusion was definitely problematic, at least coming from that starting point.

A few minutes passed, and with a rueful grin Brian passed across a few more coins. He was, happily, distracted from this embarrassing failure by the arrival of his battered companions. In the ensuing confusion he completely forgot to ask Ozwald what the solution was, and was left baffled until after dinner, when Bert figured it out for him at the price of much teasing.

What was wrong with the apparently innocent mathematical calculation?

Solution