CareerCup

This is how I would do it :

Look at it like a tree. Iterate through each element as if it is the one being chosen. Keep a boolean array to check if it is visited or not. If an element is chosen and both it's neighbors have been visited, just iterate to find either the closest left or right one (Go in Order). In case one is, visit the other. Simply do this till you hit a node whose neighbors are visited and there are no elements left [I do this with a loop every time, but you can simply keep a count of the elements too].

public class Ruffians {

	public static void main(String[] args) {
		System.out.println(houseArray(new int[]{20, 50, 50, 1, 5})); 
	}
	
	private static int houseArray(int[] houseArray) {
		return calculateBestSteal(houseArray);
	}
	
	private static int calculateBestSteal(int[] houseArray) {
		int bestSteal = 0;
		for(int startHouse = 0; startHouse < houseArray.length; startHouse++) { 
			int stealWithThisStart = stealWithStart(houseArray, startHouse); 
			bestSteal = (bestSteal < stealWithThisStart) ? stealWithThisStart : bestSteal; 
			if( stealWithThisStart > bestSteal) {
				bestSteal = stealWithThisStart;
			}
		}
        return bestSteal;		
	}
	
	private static int stealWithStart(int[] houseArray, int currentHouse) {
		boolean[] visited = new boolean[houseArray.length];
		for(int i = 0; i < visited.length; i++) {
			visited[i] = true;
		}
		boolean anyHouseLeft = true;
		int thisSteal = 0;
		while(anyHouseLeft) { 
			visited[currentHouse] = false;
			thisSteal += houseArray[currentHouse];
			if(currentHouse == 0) {
				visited[currentHouse + 1] = false;
			} else if(currentHouse == houseArray.length - 1) {
				visited[currentHouse - 1] = false; 
			} else {
				visited[currentHouse - 1] = false;
				visited[currentHouse + 1] = false;
			}			
			int nextHouse = nextHouseToChoose(currentHouse, visited);
			anyHouseLeft = (nextHouse == -1) ? false : true;
			currentHouse = nextHouse;
		}
		
		return thisSteal;
	}
	
	private static int nextHouseToChoose(int currentHouse, boolean[] visited) { 
		if(currentHouse == 0) {
			int firstTrue = -1;
			for(int visVal = 0; visVal < visited.length; visVal++) {
				if(visited[visVal]) {
					firstTrue = visVal;
				}
			}
			return firstTrue;			
		} else if(currentHouse == visited.length - 1) {
			int lastTrue = -1;
			for(int visVal = visited.length - 1; visVal >= 0; visVal--) {
				if(visited[visVal]) {
					lastTrue = visVal;
				}
			}
			return lastTrue;
		} else { 
			if(visited[currentHouse - 1]) {
				return currentHouse - 1;
			} else if(visited[currentHouse + 1]) {
				return currentHouse + 1;
			} else { 
				for(int valOnLeft = currentHouse; valOnLeft >= 0; valOnLeft--) {
					if(visited[valOnLeft]) {
						return valOnLeft;
					} 
				}
				
				for(int valOnRight = currentHouse; valOnRight < visited.length; valOnRight++) {
					if(visited[valOnRight]) {
						return valOnRight;
					} 
				}
				
				return -1;
			}
		}
	}
}