Weighings for the odd bean

You have 9090 beans that all weigh the same except one, which is either slightly heavier or slightly lighter (you don't know which). Using only a balance scale, what is the minimum number of weighings that guarantees identifying the odd bean?

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  1. Each weighing has 33 outcomes, so ww weighings give 3w3^w distinguishable results.
  2. For an odd item of unknown direction, ww weighings resolve up to (3w3)/2(3^w-3)/2 items. Find the smallest ww with that 90\ge 90.

Answer

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5

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Asked at: Jane Street, Optiver

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