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.
Do not write an equation yet. Shake the figure first.
Keep coloring smaller squares
The sum grows but never crosses one third: every colored square travels with two empty twins.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
Quarter the square and color one (¼); leave the two beside it uncolored.
Return to the exact question
Find ¼ + 1/16 + 1/64 + … using a picture that subdivides a square.
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.
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