NumberSolitaire.md

A game for one player is played on a board consisting of N consecutive squares, numbered from 0 to N − 1. There is a number written on each square. A non-empty zero-indexed array A of N integers contains the numbers written on the squares. Moreover, some squares can be marked during the game.

At the beginning of the game, there is a pebble on square number 0 and this is the only square on the board which is marked. The goal of the game is to move the pebble to square number N − 1.

During each turn we throw a six-sided die, with numbers from 1 to 6 on its faces, and consider the number K, which shows on the upper face after the die comes to rest. Then we move the pebble standing on square number I to square number I + K, providing that square number I + K exists. If square number I + K does not exist, we throw the die again until we obtain a valid move. Finally, we mark square number I + K.

After the game finishes (when the pebble is standing on square number N − 1), we calculate the result. The result of the game is the sum of the numbers written on all marked squares.

For example, given the following array:

    A[0] = 1
    A[1] = -2
    A[2] = 0
    A[3] = 9
    A[4] = -1
    A[5] = -2

one possible game could be as follows:

• the pebble is on square number 0, which is marked;
• we throw 3; the pebble moves from square number 0 to square number 3; we mark square number 3;
• we throw 5; the pebble does not move, since there is no square number 8 on the board;
• we throw 2; the pebble moves to square number 5; we mark this square and the game ends.

The marked squares are 0, 3 and 5, so the result of the game is 1 + 9 + (−2) = 8. This is the maximal possible result that can be achieved on this board.

Write a function:

function solution(A);

that, given a non-empty zero-indexed array A of N integers, returns the maximal result that can be achieved on the board represented by array A.

For example, given the array

    A[0] = 1
    A[1] = -2
    A[2] = 0
    A[3] = 9
    A[4] = -1
    A[5] = -2

the function should return 8, as explained above.

Assume that:

• N is an integer within the range [2..100,000];
• each element of array A is an integer within the range [−10,000..10,000].

Complexity:

• expected worst-case time complexity is O(N);
• expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).

Elements of input arrays can be modified.

function solution(A) {
    var n = A.length;
    var dp = new Array(n);
    
    dp[0] = A[0];

    for (var i = 1; i < n; i++) {
        var max = dp[i - 1] + A[i];

        for (var j = 1; j <= 6; j++) {
            if (i - j >= 0) {
                max = Math.max(dp[i - j] + A[i], max);
            }
        }

        dp[i] = max;
    }

    return dp[n - 1];
}