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If a=ba = b, does adding and factoring equations allow you to prove that 1=21 = 2? Flawed proofs often look completely airtight until you inspect the hidden assumptions between the lines.
Spotting mathematical fallacies is the sharpest test of logical reasoning because an argument is only as strong as its single weakest deduction.
Dividing by zero is blatant once revealed, but what happens when a proof assumes the very thing it sets out to establish?

Circular Reasoning

Circular reasoning, also called begging the question, occurs when an argument assumes its conclusion as an unstated premise.
For example, consider proving that ddx[sin⁡(x)]=cos⁡(x)\frac{d}{dx}[\sin(x)] = \cos(x) using the limit definition lim⁡h→0sin⁡(h)h=1\lim_{h \to 0} \frac{\sin(h)}{h} = 1. If you evaluate that limit using L'Hôpital's Rule, you differentiate sin⁡(h)\sin(h) into cos⁡(h)\cos(h), which requires knowing the derivative of sine before you have even proved it.
Logical loops corrupt algebra and calculus, but structural gaps can completely derail mathematical induction.

The Induction Gap

In 1961, mathematician George Pólya introduced a famous paradox claiming to prove that 'all horses have the same color' using mathematical induction.
The base case n=1n=1 is trivially true because any single horse shares its own color. The induction step assumes any group of kk horses has the same color, then considers k+1k+1 horses by overlapping two groups of kk horses.