Gauss's sum — the trick that changes how you see arithmetic

Problem: Find the sum of the first $n$ natural numbers: $1 + 2 + 3 + \cdots + n$.

Solution:

Write the sum twice — once forwards, once backwards, and add column by column:

$$S = 1 + 2 + 3 + \cdots + n$$ $$S = n + (n-1) + (n-2) + \cdots + 1$$ $$2S = (n+1) + (n+1) + \cdots + (n+1) \quad (n \text{ times})$$

Therefore:

$$S = \frac{n(n+1)}{2}$$

Why it’s beautiful: The trick is symmetry — pairing the smallest with the largest always gives $n+1$. Young Gauss reportedly computed $\sum_{k=1}^{100} k = 5050$ in seconds using exactly this idea. The geometric proof (staircase + its mirror = rectangle) is even more elegant.