¼ + 1/16 + 1/64 + … = ⅓ — Infinity at a Glance
An infinite sum that lands exactly on ⅓? Color one of four squares at each scale: each colored square always has two uncolored twins — so exactly one third.
Problem
Find ¼ + 1/16 + 1/64 + … using a picture that subdivides a square.
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.)
Quarter the square and color one (¼); leave the two beside it uncolored.
Why this approach works
Quarter the square and color one quarter (¼); subdivide a leftover and color one (1/16); repeat forever. At every scale, each colored square has exactly two same-size uncolored twins — so colored : uncolored = 1 : 2 throughout. The colored region is therefore exactly ⅓ of the whole, giving ¼ + 1/16 + 1/64 + … = ⅓.
Proof
Coloring one of four squares at every scale leaves two same-size uncolored twins per colored square. The 1:2 ratio holds at every scale, so the colored area is one third of the whole.
Try it yourself
Point at any colored square and find its two same-size uncolored twins — always two, at every scale. The 1:2 ratio never breaks, so the sum is ⅓.
Related mathematics
Beyond MathVoyage
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