https://matrix67.itch.io/this-is-math Color one of the circles in each row red, so that when the nemesis starts from the top circle and moves downwards, he can never pass through four or more red circles. While moving downwards, he can only take one step to the left or one step to the right at a time.

console.log.txt

{
  "is_find": true,
  "red_path": [
    0,
    0,
    2,
    0,
    2,
    4,
    6
  ]
}

find_red_path.js

function generate_paths(height) {

  let path = new Array(height).fill(0); //init

  const paths = [path];

  let level = height - 1,
    pos = 0; //start from bottom
  while (true) {
    pos++; //check right point

    if (pos <= path[level - 1] + 1) {
      path = JSON.parse(JSON.stringify(path)); //create new path

      for (; level < height; level++) { //go to bottom
        path[level] = pos;
      }
      level = height - 1;

      //console.log(`level:${level} ${path}`);

      paths.push(path);
    } else {
      level--; //go back
      pos = path[level];
    }

    if (pos + 1 === height) break;
  }

  return paths;
}

function random_red(height) {
  let path = [];
  for (let i = 0; i < height; i++) {
    const d = Math.floor(Math.random() * (i + 1));
    path.push(d)
  }
  //console.log(path)
  return path;
}

function generate_red_paths(height) {

  let path = new Array(height).fill(0); //init

  const paths = [path];

  let level = height - 1,
    pos = 0; //start from bottom
  while (true) {

    pos++; //check right point

    if (pos <= level) {

      path = JSON.parse(JSON.stringify(path)); //create new path
      path[level] = pos;
      for (level = level + 1; level < height; level++) { //go to bottom
        path[level] = 0;
      }
      level = height - 1;

      //console.log(`level:${level} ${path}`);

      paths.push(path);
    } else {
      if (level === 0) break;

      level--; //go back
      pos = path[level];
    }

  }

  return paths;
}

function check_pass_through(path, red) {
  let pass_through = 0;
  for (let i = 0; i < path.length; i++) {
    if (path[i] === red[i]) pass_through++;
  }
  //console.log(pass_through);
  return pass_through;
}

function find_red_path(height, max_pass_through) {
  const paths = generate_paths(height);

  let try_count = 0;
  let is_find = false;

  let red_paths = generate_red_paths(height);

  for (let red_path of red_paths) {
    is_find = true;
    for (let path of paths) {
      let pass_through = check_pass_through(path, red_path);
      if (pass_through > max_pass_through) {
        is_find = false;
        break;
      }
    }

    if (is_find) return {
      is_find,
      red_path
    };
  }


  return {
    is_find
  };

}

function find_red_path_random(height, max_pass_through, max_try_count = 1e6) {
  const paths = generate_paths(height);

  let try_count = 0;
  let is_find = false;

  let red_paths = generate_red_paths(height);

  while (++try_count < max_try_count) {
    let red_path = random_red(height);
    //console.log(red_path)
    is_find = true;
    for (let path of paths) {
      let pass_through = check_pass_through(path, red_path);
      if (pass_through > max_pass_through) {
        is_find = false;
        break;
      }
    }

    if (is_find) return {
      is_find,
      red_path
    };
  }


  return {
    is_find
  };

}

find_red_path_random(7, 3)
//or
find_red_path(7, 3)