Got this by e-mail today …
a & b are two arbit numbers and let a+b=t
(a + b)(a - b) = (a + b)(a - b)
(a + b)(a - b) = t(a - b)
a^2 - b^2 = ta - tb
a^2 - ta = b^2 - tb
a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
(a - t/2)^2 = (b - t/2)^2
a - t/2 = b - t/2
a = b
what ??? 
This step is wrong. It basically says t = a and t = b. If a=b=t, then it’s true, but t is defined as a+b.
It’s many years since I studied maths, but I don’t get the transition between:
a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
and
(a - t/2)^2 = (b - t/2)^2
Substituting with real numbers, the first line I quoted is indeed correct, but the second line I quoted is incorrect.
It’s a technique called completing the square, and is used when solving a quadratic equation. In practice, when solving something of the form ax^2 + bx = c, you take half of b and add it to both sides. That way, you can “complete the square”, getting it into the form (x + b^(1/2))^2 = c + b/2, and you can then isolate the x by taking the square root of both sides of the equation.
Note: ^(1/2) is the equivalent of taking the square root. The wiki page describes it much more succinctly.
ETA: They flubbed that line, too, though, because the extra (t^2)/4 part needs to be added to both sides of the equation,* for each time it’s done.* In this case, it was added to both sides, but the attempt to complete the square was done on both sides. It doesn’t work like that. Not to mention that it’s only done when there is a variable involved, and the problem defines a, b, and t as arbit [sic] numbers, so they’re none of them are technically variables.
The completion of the square was completely correct. What happened was that in taking square roots, the procedure extracted the negative square root on one side and the positive one on the other (except when a = b, when both sides are 0). If, for example, a > b, the a - t/2 = (a - b)/2 > 0 while b - t/2 = (b - a)/2 < 0. If a < b, the left hand side is negative and the right hand positive instead.
Mostly these things depend on division by 0, but this one is by the ambiguity of square roots.
There is a simple procedure for finding where these paradoxes run off the rail. Simply substitute a = 1 and b = 0 and see which step fails.
And here I turn up twenty minutes too late to PALATR, because Hari Seldon is exactly right.
Yep. It committed the same sin I’m always warning my SAT/ACT students against, to always keep the negative root in mind. To wit, any of these four formulations could be correct:
a - t/2 = b - t/2
a - t/2 = -(b - t/2)
-(a - t/2) = b - t/2
-(a - t/2) = -(b - t/2)
I actually think the mistake is in the first line. If a=b, then a-b=0 and we’re multiplying by that on both sides in the first line. Muliplying by zero makes everything true and is forbidden.
Tikster
Tikster, I believe “a=b” is the paradoxical conclusion, not the initial premise. I used 5 and 6 to test it, for example.