In the Largest Multiple of Three coding problem, you are given an array of digits (0-9). You need to find the largest possible number that can be formed using some or all of these digits such that the resulting number is a multiple of three. If no such number can be formed, you return an empty string. The number should not have leading zeros unless the number itself is just "0".
This is a challenging problem often seen in Microsoft and Amazon interviews because it combines several concepts: Math, Sorting, and Greedy logic. It tests your knowledge of divisibility rules—specifically that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. It also requires careful handling of edge cases, such as removing the smallest possible digits to satisfy the sum condition while keeping the overall number as large as possible.
This problem utilizes the Math and Greedy interview pattern. First, sort the digits in descending order to prepare for forming the largest number. Calculate the total sum of the digits. If the sum % 3 is 1, you must remove one digit that has a remainder of 1 (when divided by 3), or if that's not possible, two digits that have a remainder of 2. If the sum % 3 is 2, you remove one digit with remainder 2, or two digits with remainder 1.
Given digits: [8, 1, 9, 2].
Sum = 8+1+9+2 = 20.
20 % 3 = 2. To make it a multiple of 3, we need to remove a digit that gives remainder 2.
Digits with remainder 2: [8, 2].
To keep the number large, we remove the smallest one: 2.
Remaining digits: [8, 1, 9].
Sorted descending: 981.
Check: 9+8+1 = 18 (divisible by 3). The largest number is "981".
Candidates often forget the greedy aspect of removing the smallest digits to maintain the largest possible value. Another common error is not handling the case where multiple digits need to be removed (e.g., removing two '1's if no '2' is available when sum % 3 is 2). Finally, failing to handle the "all zeros" case (returning "0" instead of "000") is a frequent oversight.
Mastering "Math, Sorting, Greedy interview pattern" questions requires a solid grasp of basic number theory. Always remember the divisibility rules for 3, 9, and 11, as they are common in digit-manipulation problems. Practice sorting techniques to quickly arrange data in a way that favors a greedy approach.
| Title | Difficulty | Topics | LeetCode |
|---|---|---|---|
| Reducing Dishes | Hard | Solve | |
| Allocate Mailboxes | Hard | Solve | |
| Append K Integers With Minimal Sum | Medium | Solve | |
| Greatest Sum Divisible by Three | Medium | Solve | |
| Smallest Range II | Medium | Solve |
