Minimum Number of Days to Eat N Oranges

Minimum Number of Days to Eat N Oranges

1. What is this problem about?

In the Minimum Number of Days to Eat N Oranges interview question, you are faced with a task of consuming n oranges using three specific rules:

1. Eat one orange.
2. If n is divisible by 2, eat n/2 oranges.
3. If n is divisible by 3, eat 2*(n/3) oranges. Each action counts as one day. You want to find the minimum number of days to eat all n oranges.

2. Why is this asked in interviews?

This question is a classic example of optimization under constraints, favored by companies like Google. It tests whether a candidate can distinguish between a standard Dynamic Programming (DP) approach and a more optimized version. With n up to 2 billion, a simple O(n) DP array will fail due to memory and time limits, forcing the candidate to use recursion with memoization or a specialized BFS.

3. Algorithmic pattern used

This problem follows the Memoization, Dynamic Programming interview pattern. Because of the large input size, you must use a top-down approach with a hash map (dictionary) for memoization. The key recursive insight is: to make n divisible by 2, you might need to eat n % 2 oranges one by one first. Then you can jump to n/2. Similarly for 3. The transitions are: minDays(n) = 1 + min(n % 2 + minDays(n // 2), n % 3 + minDays(n // 3))

4. Example explanation

Suppose n = 10.

• 10 is divisible by 2. We could eat 5 (leaving 5) or eat 1 (leaving 9).
• If we eat 5, we have 5 left.
• From 9 (which is divisible by 3), we could eat 6 (leaving 3).
• By following the recursive path: 10 -> 10/2 = 5 -> 4 -> 4/2 = 2 -> 2/2 = 1 -> 0 (5 days) But with the optimized jump: 10 -> (10%2) + 10/2 = 1 + 5 = 6... Wait, the actual optimal path for 10 is: 10 -> 9 -> 3 -> 1 -> 0 (4 days). Rule 3 is very powerful!

5. Common mistakes candidates make

The most common mistake is trying to build a DP table of size n+1, which crashes for large n. Another mistake is thinking a simple greedy "always divide by 3 if possible" works—it doesn't, because sometimes eating one orange to reach a multiple of 2 or 3 is better in the long run.

6. Interview preparation tip

When you see very large input constraints (like 10^9), steer away from iterative DP. Instead, think about recursive solutions that "jump" across the state space. Practicing BFS on state graphs is also useful here, as it naturally finds the shortest path to the "zero oranges" state.