Problem
Welcome to Numistoria, a realm where the ancient game of 'The Coin Stack Conundrum' is a test of wit and precision. In this challenge, you must stack coins such that any subsequence of the stack sums to a number divisible by 3. A subsequence is formed by removing some elements of the stack without altering the order of the remaining coins. Your mission is to determine if it’s possible to arrange the given coins to meet this unique condition. Step-by-step, you will: 1. Receive an integer n, representing the number of coins. 2. Be provided with a list of n integers, each specifying the value of a coin. 3. Analyze if these coins can be stacked to ensure every subsequence's sum is divisible by 3.
Examples
- Input: n = 3, coins = [3, 6, 9]
- Output: Yes, because any subsequence (including the whole stack) will sum to a multiple of 3.
Edge case:
- Input: n = 3, coins = [1, 1, 1]
- Output: No, because any non-empty subsequence will sum to a number not divisible by 3.
Hints
- Approach: Consider using modular arithmetic to simplify the problem of checking divisibility by 3.
- Algorithm/Data Structure: A hash map could be helpful to count occurrences of remainders when each coin value is divided by 3.
- Edge Cases: Think about situations with all coins having the same value or coins with values that are multiples of 3.
- Optimization Tip: Focus on early detection of impossible combinations to avoid unnecessary calculations. Embrace the challenge and let your mathematical intuition guide you!
Input format
The first line contains an integer n (1 <= n <= 100), the number of coins. The second line contains n space-separated integers, where each integer c_i (1 <= c_i <= 1000) represents the value of a coin.
Output format
An integer representing the number of distinct sequences in which the coins can be stacked following the rule: any subsequence should have a sum divisible by 3.
Constraints
1 <= n <= 100, 1 <= c_i <= 1000
Sample input
3 1 2 3
Sample output
4