STRONG INDUCTION, MAKE SURE YOU ANSWER THE QUESTIONS$$ CORRECTLY AND IN DETAIL: We are given $n$ identical looking coins, and a scale that allows

comparing the weights of two sets of coins $($i$.$e$.,$ we can put any number

of coins on each side of the scales$).$ We know that there is exactly one

fake coin $-$ it is lighter than other coins. Prove that we can discover the

fake coin using $|$~$lo{g}_3(n)$~$|$ comparisons.

$($Example: When $n=3,$ we can discover the fake coin with one

comparison. Take any pair of coins and compare their weight: if their

weight is identical, then the third coin is the fake one; and otherwise,

one coin is lighter than the other, so it must be fake.$)$

$($Another example: When $n=6,$ we can split the coins into two sets of

three coins, and compare the weights of the two sets. What will be our

next step after the comparison?$)$