https://www.youtube.com/watch?v=RGQe8waGJ4w Comment asked what would happen if you marked 2084 as visited

Output

Normally:
Current Position: (510,477)=2084
2084 visited at the beginning:
Current Position: (474,489)=2720
Mark all blocking as visited:
Blocking numbers: 
2084,2720,3325,3753,7776,5632,7411,8562,14076,8469,9231,22702,14661,21710,21078,25809,27112,24708,19844,26943,26737,32449,31366,45036,37853,37188,43318,62095,67401,68736,70848,62789,63223,69245,85385,52467,71072,68435,76611,84206,81869,70277,81475,83776,70767,84763,99029,82609,103815,86102,93729,100614,108039,82111,99935,85283,109993,119856,119518,116066,109686,92741,124770,92378,104657,125102,107267,107246,117089,117766,99295,121575,98930,117390,123583,112565,122080,111612,111597,97349,105002,130602,133509,153410,127138,143952,153326,157774,122534,136542,163038,134778,140186,162865,171044,159637,171041,174368,184225,152988,176535,171506,147883,172360,156132,179411,179238,175850,168569,195533,191278,168006,204000,171467,144410,211291,188859,213596,225332,212859,201021,240304,233933,183242,237743,219573,162013,230993,272187,255213,254682,208965,252154,241644,245595,265411,206168,217227,263507,275293,268475,264339,233353,303280,267891,221884,226639,255175,198031,260706,297759,320601,297687,288717,239650,216254,224715,281047,315480,249064,242129,308384,329687,285288,254093,324526,330268,260691,264794,268929,273096,277295,325082,284244,270494,305377,288524,285786,357239,352551,243562,307058,360853,242079,354836,262199,372368,333134,353636,371649,372949,363210,343563,256041,312571,385237,344369,374068,419535,381448,393911,282003,268330,393969,349971,364974,350607,424145,298733,427228,288377,300326,292687,292171,448411,324947,463354,283571,287850,438460,464628,385823,305817,467422,495930,332415,345209,352902,353524,337607,345791,502996,483451,494488,509364,374614,424571,429802,507215,516509,537602,370924,558347,471453,552362,573389,563589,562883,446364,398237,421306,378260,381944,382608,387575,579461,567976,453058,400752,574949,443675,425181,375788,513558,402664,409018,423245,454392,590121,467984,375181,605614,422553,567963,453038,605590,571789,422533,360606,541125,463873,634736,525072,448974,427749,528720,449664,643522,583942,461729,459044,554467,472743,485924,653258,459079,654825,555231,489398,677697,423813,451620,521407,671820,517577,697598,606277,446953,558569,694257,637842,463803,612514,593121,517781,624340,653915,575499,581586,715200,555910,735686,549948,669390,736507,538034,557402,593055,684965,677620,669371,697533,597681,591505,590756,801426,784455,711722,525679,586033,730432,575456,796030,515514,670965,712602,728684,785320,680829,833977,598409,603858,631407,753718,606202,642547,710871,850506,850479,829375,525646,519856,525643,578459,590623,645750,863450,860620,874637,852300,887772,867148,751917,621030,835720,767643,832063,817524,750199,884908,882106,832057,816560,913368,712510,602201,779932,697399,796235,771129,880188,726081,620981,576116,582207,643273,727800,695677,816875,949086,848585,711560,884857,795909,608405,813867,637609,806654,679075,685686,665056,641676,648103,767577,822891,942246,670862,678176,933516,600655,841136,682350,709961,739738,976550,736293,971526,628893,934452,713323,899933,798533,738023,777142,704783,859574,1028631,739454,696429,846637,854016,675718,1005398,757031,768429,821041,949013,1031644,919000,955795,1028594,1052072,830140,1081025,813778,793169,737075,842925,1022520,822847,800875,854933,880066,736221,1069581,952890,1078893,1087220,945070,797585,948960,811957,1108193,899903,1049983,934388,757736,864083,807182,1025479,736199,743082,972482,760431,1007291,912259,1152838,823611,1010321,1134671,856636,1166879,882770,890305,1126279,1022444,770926,852076,1199517,907478,854565,980366,1066376

TrappedKnight.java

import java.util.*;

public class TrappedKnight {

	public enum Direction {
		LEFT,RIGHT,UP,DOWN
	}

	public HashSet<Integer> visited;
	int[][] matrix;
	int size; //Size of matrix
	public int xpos, ypos;

	public TrappedKnight (int size) {
		this.size = size;
		xpos = size/2;
		ypos = size/2;
		visited = new HashSet<>();
	}

	public static void main(String[] args) {
		System.out.println("Normally:");
		TrappedKnight tk = new TrappedKnight(1000);
		tk.preCompute();
		tk.makeStepUntilFail();		
		System.out.println("Current Position: ("+tk.xpos+","+tk.ypos+")="+tk.matrix[tk.xpos][tk.ypos]);

		System.out.println("2084 visited at the beginning:");
		TrappedKnight tk2 = new TrappedKnight(1000);
		tk2.preCompute();
		tk2.visited.add(Integer.valueOf(2084));
		tk2.makeStepUntilFail();	
		System.out.println("Current Position: ("+tk2.xpos+","+tk2.ypos+")="+tk2.matrix[tk2.xpos][tk2.ypos]);

		System.out.println("Mark all blocking as visited:");
		TrappedKnight tk3 = new TrappedKnight(10000);
		tk3.preCompute();
		HashSet<Integer> visitedUpdated = new HashSet<>();
		System.out.println("Blocking numbers: ");
		while(true) {
			tk3.xpos = tk3.size/2;
			tk3.ypos = tk3.size/2;
			int ret = tk3.makeStepUntilFail();
			visitedUpdated.add(Integer.valueOf(ret));
			tk3.visited = new HashSet<>(visitedUpdated);
			System.out.print(ret+",");
			//try { System.in.read(); } catch(Exception e) { }
		}
	}

	public int makeStepUntilFail() {
		int i = 1;
		visited.add(Integer.valueOf(1));
		boolean step = true;
		while (step) {
			//drawOutput(10);
			step = makeStep();
			//try { System.in.read(); } catch(Exception e) { }
		}
		return matrix[xpos][ypos];
	}

	public int getMinIndex(int[] arr) {
		int min = Integer.MAX_VALUE;
		int ret = -1;
		for (int i = 0; i< arr.length; i++) {
			if (arr[i]<min) {
				min = arr[i];
				ret = i;
			}
		}
		return ret;
	}

	public int[] getStep() {
		int[] choices = new int[8];
		choices[0] = matrix[xpos-1][ypos+2];
		choices[1] = matrix[xpos+1][ypos+2];
		choices[2] = matrix[xpos+2][ypos-1];
		choices[3] = matrix[xpos+2][ypos+1];
		choices[4] = matrix[xpos-1][ypos-2];
		choices[5] = matrix[xpos+1][ypos-2];
		choices[6] = matrix[xpos-2][ypos-1];
		choices[7] = matrix[xpos-2][ypos+1];

		for (int i = 0; i< 8; i++) {
			if (visited.contains(Integer.valueOf(choices[i]))) {
				choices[i] = Integer.MAX_VALUE;
			}
		}


		int minIndex = getMinIndex(choices);
		if (minIndex==-1) {
			return new int[]{-1,-1};
		}
		switch (minIndex) {
			case 0:
				return new int[]{xpos -1,ypos +2};
			case 1:
				return new int[]{xpos +1,ypos +2};
			case 2:
				return new int[]{xpos +2,ypos -1};
			case 3:
				return new int[]{xpos +2,ypos +1};
			case 4:
				return new int[]{xpos -1,ypos -2};
			case 5:
				return new int[]{xpos +1,ypos -2};
			case 6:
				return new int[]{xpos -2,ypos -1};
			case 7:
				return new int[]{xpos -2,ypos +1};
		}
		System.out.println(minIndex+"WHAT WHY");
		return new int[]{};
	}

	public boolean makeStep() {
		int[] step = getStep();
		if (step[0]==-1) {
			return false;
		}
		xpos = step[0];
		ypos = step[1];
		visited.add(Integer.valueOf(matrix[xpos][ypos]));
		return true;
	}

	public void drawOutput(int window) {
		for (int y = ypos-window; y<ypos+window; y++) {
			for (int x = xpos-window; x<xpos+window; x++) {
				if (xpos == x && ypos == y) {
					System.out.print(">"+matrix[x][y]+"<\t");
				} else {
					System.out.print(matrix[x][y]+"\t");
				}
			}
			System.out.println();
		}
	}

	public void preCompute() {
		int x = size/2;
		int y = size/2;
		matrix = new int[size][size];
		Direction cur = Direction.RIGHT;
		int goForXSteps = 1;
		boolean first = true;
		int remainingSteps = 1;
		for (int i = 1; x<size && y < size; i++) {
			matrix[x][y] = i;

			if (remainingSteps==0) {
				if (first) {
					first = false;
				} else {
					first = true;
					goForXSteps +=1;
				}
				remainingSteps = goForXSteps;
				switch (cur) {
					case RIGHT:
						cur = Direction.UP;
						break;
					case DOWN:
						cur = Direction.RIGHT;
						break;
					case LEFT:
						cur = Direction.DOWN;
						break;
					case UP:
						cur = Direction.LEFT;
						break;
				}
			}

			switch (cur) {
				case RIGHT:
					remainingSteps--;
					x++;
					break;
				case DOWN:
					remainingSteps--;
					y++;
					break;
				case LEFT:
					remainingSteps--;
					x--;
					break;
				case UP:
					remainingSteps--;
					y--;
					break;
			}
		}
	}
}