Triangular Numbers: 1+2+…+n = n(n+1)/2
A fast way to add 1+2+…+n? Stack it as a staircase, mate an upside-down copy, and you get a rectangle — the answer is half.
Problem
Find 1 + 2 + 3 + … + n (a triangular number) by mating staircases into a rectangle.
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
Stack 1+2+3+4+5 as a staircase (triangular number T₅).
Why this approach works
Stack 1+2+…+n as a staircase. Make an identical copy, rotate it 180°, and mate it on top: the two jagged staircases lock into a perfect n by (n+1) rectangle. Two staircases fill n(n+1), so one is half — 1+2+…+n = n(n+1)/2. The geometric form of young Gauss’s pairing trick (1+100, 2+99, …).
Proof
A staircase and its 180°-rotated copy mate into an n×(n+1) rectangle. Two staircases make one rectangle, so 1+2+…+n = n(n+1)/2.
Try it yourself
For n=100 the rectangle is 100×101, half of which is 5050 — exactly Gauss’s mental answer.
Related mathematics
Beyond MathVoyage
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