# Coin Game

Posted: 3 Dec, 2020
Difficulty: Easy

## PROBLEM STATEMENT

#### You are given an array of integers, say, ‘ARR’ of size ‘N’ containing the values associated with ‘N’ coins. Your task is to determine the maximum value of coins you can obtain by following two rules:

``````a) Both players play in alternate turns and they can remove only one coin in their turn.
b) Any player can remove coins only from either of the two ends of ‘ARR’.
``````

#### Note:

``````There can be more than one set of coins with maximum value.
``````

#### For example:

``````a) Consider 3 coins are placed with values [10, 20, 30]: Wong removes 30, then Strange removes 20, then Wong removes 10. Now all coins are taken, and Wong has coins with value 40 and he wins.

b) Consider 1 coin is placed with value [100]: Wong removes the coin and no other coin is left. So, Wong wins with value 100.
``````

#### Note:

``````a) The game only ends when NO MORE COIN IS LEFT to play with.

b) If a game ends in a draw, Wong is declared the winner.
``````
##### Input Format :
``````The first line of input contains an integer 'T', which denotes the number of the test cases. Then the test case follows.

The first line of every test case contains an integer ‘N’ representing the size of the array.

The second line of every test case contains ‘N’ single space-separated integers representing the array elements.
``````
##### Output Format :
``````For each test case, print a single integer representing the maximum value of coins you can get for a winning case.

Print the output of each test case in a separate line.
``````
##### Note:
``````You do not need to print anything; it has already been taken care of. Just implement the given function.
``````
##### Constraints:
``````1 <= T <= 5
1 <= N <= 10 ^ 3
1 <= ARR[i] <= 10 ^ 7

Where ‘ARR[ i ]’ denotes the value for ‘ith’ element of the array ‘ARR’.

Time Limit: 1 sec.
``````
Approach 1

The idea is to check each element from the array ‘ARR’ recursively whether including it in the result will give us the maximum sum or not.

Approach :

First, we will define a recursive function, say ‘MAXCOINS’ that accepts as parameters, an array of integers ‘ARR’, iterator pointer ‘STARTINDEX’ which will iterate throughout the ‘ARR’,  iterator pointer ‘ENDINDEX’ which will iterate from  the last element of ‘ARR’ till ‘STARTINDEX’ + 1 and ‘SUM’ which will contain the sum of the values of ‘ARR’ and do:

• Check if ‘ENDINDEX’ = ‘STARTINDEX’ + 1 i.e. only two elements are left. Return maximum of ‘ARR[STARTINDEX]’ and ‘ARR[ENDINDEX]’.
• Case1 - when you consider picking up the coin from the first index. Call function ‘MAXCOINS’ for ‘ARR’, ‘STARTINDEX’ + 1, ‘ENDINDEX’, and ‘SUM’  - ‘ARR[STARTINDEX]’ and subtract the value returned by it from SUM, say the value obtained is stored in a variable ‘VAL1’.
• Case2 - when you consider picking up the coin from the end index. Call function ‘MAXCOINS’ for ‘ARR’, ‘STARTINDEX’,  ‘ENDINDEX’ - 1 and ‘SUM’ - ‘ARR[ENDINDEX]’ and subtract the value returned by it from ‘SUM’, say value obtained is stored in a variable ‘VAL2’.
• Return MAX of VAL1 and VAL2.