25 Points

using System;
using System.Linq;
using System.Collections.Generic;

namespace _25points;

public static class Program {
  public static void Main() {
    int size = 9, size2 = size * size;
    int X(int n) => n / size;
    int Y(int n) => n % size;
    int cnt;
    int max = size * 2;
    var p = new int[max];
    void Show() {
      for (int i = 0; i < cnt; i++)
        Console.Write($" ({X(p[i])};{Y(p[i])})");
    }
    void Show1() {
      Console.WriteLine();
      for (int i = 0; i < size2; i++) {

        if (Array.IndexOf(p, i) >= 0) {
          Console.ForegroundColor = ConsoleColor.Red;
          Console.Write(" •");
        } else {
          Console.ForegroundColor = ConsoleColor.Yellow;
          Console.Write(" •");
        }
        if ((i + 1) % size == 0) Console.Write("\n");
      }
      Console.ForegroundColor = ConsoleColor.White;
    }
    bool Test() {
      if (cnt < 3) return true;
      double x = X(p[cnt - 1]), y = Y(p[cnt - 1]);
      for (int i = 0; i < cnt - 2; i++) {
        double x1 = X(p[i]), y1 = Y(p[i]);
        for (int j = i + 1; j < cnt - 1; j++) {
          double x2 = X(p[j]), y2 = Y(p[j]);
          if ((x - x1) * (y - y2) == (x - x2) * (y - y1)) return false;
        }
      }
      return true;
    }
    bool[,] F() {
      var f = new bool[size, size];
      for (int i = 0; i < max; i++) {
        f[X(p[i]), Y(p[i])] = true;
      }
      return f;
    }
    bool Eq(bool[,] f0, bool[,] f) {
      bool[,] Copy(bool[,] f) {
        var ff = new bool[size, size];
        for (int i = 0; i < size; i++)
          for (int j = 0; j < size; j++)
            ff[i, j] = f[i, j];
        return ff;
      }
      bool[,] Op1(bool[,] f) {
        var ff = new bool[size, size];
        for (int i = 0; i < size; i++)
          for (int j = 0; j < size; j++)
            ff[size - 1 - i, j] = f[i, j];
        return ff;
      }
      bool[,] Op2(bool[,] f) {
        var ff = new bool[size, size];
        for (int i = 0; i < size; i++)
          for (int j = 0; j < size; j++)
            ff[j, i] = f[i, j];
        return ff;
      }
      var ff = Copy(f0);
      for (int i = 0; i < 8; i++) {
        bool eq = true;
        ff = ((i & 1) == 0 ? Op1(ff) : Op2(ff));
        for (int j = 0; j < size; j++)
          for (int k = 0; k < size; k++)
            if (ff[j, k] != f[j, k]) eq = false;
        if (eq) return true;
      }
      return false;
    }
    var l = new List<bool[,]>();
    cnt = 1;
    int cnt1 = 0;
    p[0] = 0;
    bool tst = true;
    while (cnt > 0) {
      if (tst) {
        //Console.Clear();
        //Show();
        if (cnt == max) {
          //Console.Clear();
          //Console.WriteLine("OK" + (++cnt1));
          bool[,] f = F();
          bool test1 = true;
          for (int i = 0; i < l.Count; i++)
            if (Eq(l[i], f)) { test1 = false; break; }
          if (test1) {
            Show1();
            l.Add(f);
          }
          //return;
          //Console.ReadLine();
          tst = false;
        } else {
          if (p[cnt - 1] < size2 - 1) {
            cnt++;
            p[cnt - 1] = p[cnt - 2] + 1;
            tst = Test();
          } else {
            cnt--;
            tst = false;
          }
        }
      } else {
        if (p[cnt - 1] < size2 - 1) {
          p[cnt - 1]++;
          tst = Test();
        } else cnt--;
      }

    }
    Console.WriteLine(l.Count);
  }
}