Brain Teasers · The Infinite Power Tower
Infinite Power Tower Puzzle - Play It (x = Root 2)
x to the x to the x, forever, equals 2. Find x without a calculator.
What is an infinite power tower?
The tower x^(x^(x^...)) is the limit of x, x^x, x^(x^x) and so on. Like a continued fraction, it contains a copy of itself.
Writing it as an equation in its own value removes the infinite part, but you still have to check the limit exists.
How it works
A positive x stacked as an infinite power tower equals 2. The story asks for x. You type your answer and check it. Then you step through the answer and the breakdown.
How scoring works
This teaser checks your answer. Any correct form counts - a fraction, a decimal rounded correctly, a percent, or the number in words. You get two tries before the answer opens. A correct first try earns a ticket; a second try or opening the answer early is recorded but earns nothing. The story teasers share one lifetime unlock for free users.
The key insight
The exponent above the bottom x is the whole tower again, so x^2 = 2 and x = √2. Its tower climbs 1.414, 1.633, 1.761, ... toward 2 and never passes it.
The full step-by-step solution lives at /brain-teasers/infinite-power-tower/solution.
A worked example
If the tower 'equals 4', the same move gives x^4 = 4, so x = √2 again - but √2's tower converges to 2, so no x gives 4.
Common mistakes
• Answering 2, or solving x^x = 2.
• Skipping the convergence check.
• Missing the tower-equals-4 trap in the follow-up.
Why interviews test this
Interviewers want the self-reference step fast, then a word about convergence. The written solution is at /brain-teasers/infinite-power-tower/solution.
Play The Infinite Power Tower on Outcry
The Infinite Power Tower is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play The Infinite Power Tower and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.