Как написать алгоритм для прохода конем и ладьей всей шахматной доски (от 3на3 до 10на10) без повторов?

const knightMoves = [[-2, -1], [-1, -2], [1, -2], [2, -1], [2, 1], [1, 2], [-1, 2], [-2, 1]];
const checkBorders = (x, y) => x >= 0 && x < size && y >= 0 && y < size;

let size = 8;
let numFields = size * size;
let startIndex = 0;
let route = []; // маршрут
let chess = []; // поля доски

drawBoard();

// Поиск пути
function knightTour() {

  // Досрочный ответ
  if (size % 2 == 1 && startIndex % 2 == 1)
    return alert('Путь не найден!');

  // Подготовка поля
  chess = [];

  for (let i = 0; i < size; i++) {
    for (let j = 0; j < size; j++) {

      let idx = i * size + j;
      let adj = [];

      for (let m of knightMoves) {
        let aj = j + m[0],
          ai = i + m[1];

        if (checkBorders(aj, ai)) adj.push(ai * size + aj);
      };

      chess.push({
        visited: false,
        adj: adj
      });
    }
  }

  // Поиск
  route = [];

  if (search(startIndex)) showRoute(route);
  else alert('Путь не найден!');
}

// DFS с сортировкой Варнсдорфа
function search(v, lvl = 1) {
  route.push(v);

  if (lvl == numFields) return true;
  if (chess[v].visited) return false;

  chess[v].visited = true;
  chess[v].adj.sort((a, b) => movesCount(a) - movesCount(b));

  for (let neighbor of chess[v].adj) {
    if (!chess[neighbor].visited) {
      if (search(neighbor, lvl + 1)) return true;
    }
  }

  chess[v].visited = false;
  route.pop();

  return false;
}

// Количество доступных ходов из поля idx
function movesCount(idx) {
  let count = 0;
  let adj = chess[idx].adj;

  for (let i = 0; i < adj.length; i++) {
    count += +!chess[adj[i]].visited;
  }
  return count;
}




////-----------
///
// DOM-события
wrap.onclick = (e) => {
  hideRoute();

  if (e.target.nodeName == 'DIV')
    setStartIndex(e.target.getAttribute('idx'));
}

boardsize.onchange = (e) => {
  hideRoute();
  size = e.target.value;
  numFields = size * size;
  drawBoard();
}


// Вспомогалки DOM-игралки
function drawBoard() {
  chessboard.textContent = '';
  for (let i = 0; i < numFields; i++) appendField(size, i);
  setStartIndex(0);
}


function setStartIndex(idx) {
  chessboard.children[startIndex].classList.remove('first');
  chessboard.children[idx].classList.add('first');
  chessboard.children[idx].innerHTML = '&#9822;';
  startIndex = idx;
}

function showRoute(route) {
  let d = '';
  let k = chessboard.offsetWidth / size;
  let l = route.length - 1;

  route.forEach((v, i) => {
    chessboard.children[v].innerHTML = (i == l ? '&#9822' : i + 1);

    let y = (~~(v / size) + 0.5) * k;
    let x = ((v % size) + 0.5) * k;

    d += (i ? (i == 1 ? ' L' : ' ') : ' M') + x + ',' + y;
  });

  path.setAttribute('d', d);
  chessboard.classList.remove('mute');
  drawing.style.display = 'block';
}

function hideRoute() {
  drawing.style.display = 'none';
  chessboard.classList.add('mute');
}


function appendField(size, idx) {
  let div = document.createElement('div');

  let q = size % 2 ? 0 : ~~(idx / size) % 2;
  if ((idx + q) % 2) div.className = 'dark';

  div.setAttribute('idx', idx);
  div.style.width = div.style.height = (100 / size).toFixed(3) + '%';
  chessboard.append(div);
}

#wrap {
  width: 400px;
  height: 400px;
  position: relative;
}

#chessboard {
  display: flex;
  flex-wrap: wrap;
  width: 100%;
  height: 100%;
  margin-top: 1em;
}

#chessboard div {
  display: flex;
  box-sizing: border-box;
  position: relative;
  background: #fece9e;
  justify-content: center;
  align-items: center;
  font-size: 1em;
  cursor: pointer;
}

#chessboard div:hover { border: 2px solid red; }
#chessboard div.dark { background: #d08c47; }

#drawing {
  position: absolute;
  width: 100%;
  height: 100%;
  z-index: 0;
  top: 0;
  left: 0;
  display: none;
}

.mute { color: rgba(255, 255, 255, 0); }
#chessboard div.first { background: yellow; color: green; }
#chessboard div.last { background: blue; }

Размер: <input id="boardsize" name="size" type="number" min="3" max="50" value="8" />
<button onclick="knightTour()">Найти</button>
<div id="wrap">
  <div id="chessboard"></div>
  <svg id="drawing"><path id="path" d="" stroke="blue" fill="none" /></svg>
</div>

const rookMoves = [[1, 0], [0, 1], [-1, 0], [0, -1]];
const knightMoves = [[-2, -1], [-1, -2], [1, -2], [2, -1], [2, 1], [1, 2], [-1, 2], [-2, 1]];

const checkBorders = (x, y) => x >= 0 && x < size && y >= 0 && y < size;
chessman.onchange = (e) => mode.disabled = chessman.value === 'knightTour';

let size = 8;
let numFields = size * size;
let startIndex = 0;
let route = []; // маршрут
let chess = []; // доска
let wtmat = []; // доп матрица весов
let useWtmat = true;

drawBoard();

// Инициализация доски
function prepareBoard(moves) {
  chess = [];

  for (let i = 0; i < size; i++) {
    for (let j = 0; j < size; j++) {
      let idx = i * size + j;
      let adj = [];

      for (let m of moves) {
        let aj = j + m[0], ai = i + m[1];
        if (checkBorders(aj, ai)) adj.push(ai * size + aj);
      };

      chessboard.children[idx].innerHTML = '';
      chess.push({ visited: false, adj: adj, rank: adj.length });
    }
  }

  wtmat = getWtmat(size);
}

// Поиск пути для ладьи
function rookTour() {
  // Досрочный ответ
  if (size % 2 == 1 && startIndex % 2 == 1)
    return alert('Путь не найден!');

  useWtmat = !!+mode.value;

  let bestRoute = [];
  let n = 0;

  while (n++ < 8) {
    if (n % 5 == 0) rookMoves.reverse();

    prepareBoard(rookMoves);
    route = [];

    if (search(startIndex, 2 - mode.value)) {
      simplifyRoute();
      if (bestRoute.length == 0) bestRoute = route.slice();
      else if (route.length < bestRoute.length) bestRoute = route.slice();
    }

    rookMoves.push(rookMoves.shift());
  }

  if (bestRoute.length)
    showRoute(bestRoute);
  else
    alert('Путь не найден!');
}

// Поиск пути для коня
function knightTour() {
  // Досрочный ответ
  if (size % 2 == 1 && startIndex % 2 == 1)
    return alert('Путь не найден!');

  useWtmat = true;
  let n = 0, success = false;

  while (!success && n++ < 16) {
    if (n % 9 == 0) knightMoves.reverse();

    prepareBoard(knightMoves);
    route = [];

    success = search(startIndex, 1)
    if (!success) knightMoves.push(knightMoves.shift());
  }

  if (success)
    showRoute(route);
  else
    alert('Путь не найден!');
}

// Обход доски без возвратов
function search(v, dr = 1) {
  route.push(v);

  for (let lvl = 1; lvl < numFields; lvl++) {
    chess[v].visited = true;
    changeRank(v, -dr);
    chess[v].adj.sort(sortMoves);

    let next = chess[v].adj.findIndex(i => !chess[i].visited);

    if (next == -1) {
      resetBoard();
      route = [];
      return false;
    }

    v = chess[v].adj[next];
    route.push(v);
  }

  return true;
}

function resetBoard() {
  for (let i = 0; i < numFields; i++) {
    chess[i].visited = false;
    chess[i].rank = chess[i].adj.length;
  }
}

function changeRank(v, r) {
  chess[v].rank += r;
  chess[v].adj.forEach(q => chess[q].rank += r);
}

function sortMoves(a, b) {
  let ma = chess[a].rank;
  let mb = chess[b].rank;

  if (useWtmat && ma == mb) return wtmat[a] - wtmat[b];
  else
    return ma - mb;
}

// Фильтруем лишние шаги для ладьи
function simplifyRoute() {
  let out = [];
  let prev = 0;

  for (i = 0; i < route.length - 1; i++) {
    let amb = Math.abs(route[i] - route[i + 1]);
    if (amb != prev) out.push(route[i]);
    prev = amb;
  }

  out.push(route[i]);
  route = out;
}

// Дополнителная матрица весов
function getWtmat(size) {
  let mat = Array(size * size).fill(0);
  let o = 0;
  let s = size;

  while (s > 0) {
    for (let i = 0; i < s; i++) {
      let q = o + s - 1;
      let r = (o + i);

      mat[o * size + r] = o;
      mat[q * size + r] = o;
      mat[r * size + o] = o;
      mat[r * size + q] = o;
    }
    s -= 2;
    o++;
  }

  return mat;
}




////-----------
///
// DOM-события
wrap.onclick = (e) => {
  hideRoute();

  if (e.target.nodeName == 'DIV')
    setStartIndex(e.target.getAttribute('idx'));
}

boardsize.onchange = (e) => {
  hideRoute();
  size = e.target.value;
  numFields = size * size;
  drawBoard();
}


// Вспомогалки DOM-игралки
function drawBoard() {
  chessboard.textContent = '';
  for (let i = 0; i < numFields; i++) appendField(size, i);
  setStartIndex(0);
}


function setStartIndex(idx) {
  if (startIndex < numFields)
    chessboard.children[startIndex].classList.remove('first');
  chessboard.children[idx].classList.add('first');
  chessboard.children[idx].innerHTML = '&#9822;';
  startIndex = idx;
}

function showRoute(route) {
  let d = '';
  let k = chessboard.offsetWidth / size;
  let l = route.length - 1;

  route.forEach((v, i) => {
    chessboard.children[v].innerHTML = (i == l ? '&#9822' : i + 1);

    let y = (~~(v / size) + 0.5) * k;
    let x = ((v % size) + 0.5) * k;

    d += (i ? (i == 1 ? ' L' : ' ') : ' M') + x + ',' + y;
  });

  path.setAttribute('d', d);
  chessboard.classList.remove('mute');
  drawing.style.display = 'block';
}

function hideRoute() {
  drawing.style.display = 'none';
  chessboard.classList.add('mute');
}


function appendField(size, idx) {
  let div = document.createElement('div');

  let q = size % 2 ? 0 : ~~(idx / size) % 2;
  if ((idx + q) % 2) div.className = 'dark';

  div.setAttribute('idx', idx);
  div.style.width = div.style.height = (100 / size).toFixed(3) + '%';
  chessboard.append(div);
}

#wrap {
  width: 400px;
  height: 400px;
  position: relative;
}

#chessboard {
  display: flex;
  flex-wrap: wrap;
  width: 100%;
  height: 100%;
  margin-top: 1em;
}

#chessboard div {
  display: flex;
  box-sizing: border-box;
  position: relative;
  background: #fece9e;
  justify-content: center;
  align-items: center;
  font-size: 1em;
  cursor: pointer;
}

#chessboard div:hover { border: 2px solid red; }
#chessboard div.dark { background: #d08c47; }

#drawing {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  left: 0;
  display: none;
}

.mute {
  color: rgba(255, 255, 255, 0);
}

#chessboard div.first { background: yellow; color: green; }
#chessboard div.last { background: blue; }

Размер: <input id="boardsize" name="size" type="number" min="3" max="100" value="8" />
<select id="chessman">
  <option value="rookTour" selected>Ладья</option>
  <option value="knightTour">Конь</option>
</select>
<select id="mode">
  <option value="1" selected>Спираль</option>
  <option value="0">Зиг-заг</option>
</select>
<button onclick="window[chessman.value]()">Найти</button>
<div id="wrap">
  <div id="chessboard"></div>
  <svg id="drawing"><path id="path" d="" stroke="blue" fill="none" /></svg>
</div>