If the difference between two numbers is 6, and the difference of their squares is 180, what is the sum of the numbers?

This puzzle can be solved surprisingly quickly by using a classic algebraic shortcut known as the "difference of squares." This fundamental identity states that the difference between two squared numbers is equal to the product of their sum and their difference. In algebraic terms, this is written as (x² - y²) = (x + y)(x - y). We are given that the difference of the squares is 180 and the difference of the numbers is 6. By plugging these values into the formula, we get 180 = (the sum of the numbers) * 6. A simple division, 180 divided by 6, reveals that the sum must be 30.

With the sum (30) and the difference (6) now known, it's easy to identify the two numbers themselves as 18 and 12. We can double-check the math to be certain. The difference between 18 and 12 is indeed 6. Squaring each number gives us 324 (for 18) and 144 (for 12). The difference between these squares is 324 - 144, which equals 180, confirming the solution is correct.

This problem is a perfect illustration of how recognizing an underlying mathematical pattern can transform a tricky-looking puzzle into a straightforward calculation. The difference of squares identity is a cornerstone of algebra, used for everything from factoring polynomials to simplifying complex expressions. It demonstrates how a little bit of foundational knowledge can provide an elegant and efficient path to the right answer, bypassing more complicated guesswork or trial and error.