Eighty-One Coins, One Light

Split into three equal groups each time.

$3^4=81$.

1. 1. Lower bound

   $3^3=27<81$, so three weighings are not enough.

2. 2. Strategy

   Weigh $27$ vs $27$; the light group (or the unweighed one) has $27$ coins. Repeat with $9$, $3$, $1$: four weighings.

3. 3. Key idea

   With a known direction, $n$ coins need $\lceil {\log }_{3}n\rceil$ weighings.

Answer: 4