AdventOfCode2023/solvers/UNeverTellMeTheOdds.pas

{
  Solutions to the Advent Of Code.
  Copyright (C) 2023-2024  Stefan Müller

  This program is free software: you can redistribute it and/or modify it under
  the terms of the GNU General Public License as published by the Free Software
  Foundation, either version 3 of the License, or (at your option) any later
  version.

  This program is distributed in the hope that it will be useful, but WITHOUT
  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
  FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.

  You should have received a copy of the GNU General Public License along with
  this program.  If not, see <http://www.gnu.org/licenses/>.
}

unit UNeverTellMeTheOdds;

{$mode ObjFPC}{$H+}

interface

uses
  Classes, SysUtils, Generics.Collections, Math, matrix, USolver, UNumberTheory, UBigInt;

type

  { THailstone }

  THailstone = class
  public
    Position, Velocity: Tvector3_extended;
    constructor Create(const ALine: string);
    constructor Create(const APosition, AVelocity: Tvector3_extended);
  end;

  THailstones = specialize TObjectList<THailstone>;

  { TFirstCollisionPolynomial }

  TFirstCollisionPolynomial = class
  private
    FA: array[0..10] of TBigInt;
    FH: array[0..6] of TBigInt;
    procedure NormalizeCoefficients;
  public
    procedure Init(constref AHailstone1, AHailstone2, AHailstone3: THailstone; const t_0, t_1, t_2: Int64);
    function EvaluateAt(const AT0: Int64): TBigInt;
    function CalcPositiveIntegerRoot: Int64;
    function CalcT1(const AT0: Int64): Int64;
  end;

  { TNeverTellMeTheOdds }

  TNeverTellMeTheOdds = class(TSolver)
  private
    FMin, FMax: Int64;
    FHailstones: THailstones;
    FA: array[0..10] of Int64;
    FH: array[0..6] of Int64;
    function AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
    procedure FindRockThrow(const AIndex1, AIndex2, AIndex3: Integer);
  public
    constructor Create(const AMin: Int64 = 200000000000000; const AMax: Int64 = 400000000000000);
    destructor Destroy; override;
    procedure ProcessDataLine(const ALine: string); override;
    procedure Finish; override;
    function GetDataFileName: string; override;
    function GetPuzzleName: string; override;
  end;

const
  CIterationThreshold = 0.00001;
  CEpsilon = 0.0000000001;

implementation

{ THailstone }

constructor THailstone.Create(const ALine: string);
var
  split: TStringArray;
begin
  split := ALine.Split([',', '@']);
  Position.init(
    StrToFloat(Trim(split[0])),
    StrToFloat(Trim(split[1])),
    StrToFloat(Trim(split[2])));
  Velocity.init(
    StrToFloat(Trim(split[3])),
    StrToFloat(Trim(split[4])),
    StrToFloat(Trim(split[5])));
end;

constructor THailstone.Create(const APosition, AVelocity: Tvector3_extended);
begin
  Position := APosition;
  Velocity := AVelocity;
end;

{ TFirstCollisionPolynomial }

procedure TFirstCollisionPolynomial.NormalizeCoefficients;
var
  shift: Integer;
  i: Low(FA)..High(FA);
  //gcd: TBigInt;
begin
  // Eliminates zero constant term.
  shift := 0;
  while (shift <= High(FA)) and (FA[shift] = 0) do
    Inc(shift);

  if shift <= High(FA) then
  begin
    if shift > 0 then
    begin
      for i := Low(FA) to High(FA) - shift do
        FA[i] := FA[i + shift];
      for i := High(FA) - shift + 1 to High(FA) do
        FA[i] := 0;
    end;

    //// Finds GCD of all coefficients.
    //gcd := FA[Low(FA)];
    //for i := Low(FA) + 1 to High(FA) do
    //  if FA[i] <> 0 then
    //    gcd := TNumberTheory.GreatestCommonDivisor(gcd, FA[i]);
    //WriteLn('GCD: ', gcd);
    //
    //for i := Low(FA) to High(FA) do
    //  FA[i] := FA[i] div gcd;
  end;

  //WriteLn('(', FA[10], ') * x^10 + (', FA[9], ') * x^9 + (', FA[8], ') * x^8 + (', FA[7], ') * x^7 + (',
  //  FA[6], ') * x^6 + (', FA[5], ') * x^5 + (', FA[4], ') * x^4 + (', FA[3], ') * x^3 + (', FA[2], ') * x^2 + (',
  //  FA[1], ') * x + (', FA[0], ')');
end;

procedure TFirstCollisionPolynomial.Init(constref AHailstone1, AHailstone2, AHailstone3: THailstone; const t_0, t_1,
  t_2: Int64);
var
  P_00, P_01, P_02, P_10, P_11, P_12, P_20, P_21, P_22,
  V_00, V_01, V_02, V_10, V_11, V_12, V_20, V_21, V_22: Int64;
  k: array[0..139] of TBigInt;
  // For debug calculations
  act, a_1, a_2, b_0, b_1, c_0, c_1, d_0, d_1, e_0, e_1, f_0, f_1, f_2: Int64;
begin
  // Solving this non-linear equation system, with velocities V_i and start positions P_i:
  //   V_0 * t_0 + P_0 = V_x * t_0 + P_x
  //   V_1 * t_1 + P_1 = V_x * t_1 + P_x
  //   V_2 * t_2 + P_2 = V_x * t_2 + P_x
  // Which gives:
  //   P_x = (V_0 - V_x) * t_0 + P_0
  //   V_x = (V_0 * t_0 - V_1 * t_1 + P_0 - P_1) / (t_0 - t_1)
  // And with vertex components:
  //   1: 0 = (t_1 - t_0) * (V_00 * t_0 - V_20 * t_2 + P_00 - P_20) - (t_2 - t_0) * (V_00 * t_0 - V_10 * t_1 + P_00 - P_10)
  //   2: t_1 = (((V_01 - V_21) * t_2 + P_11 - P_21) * t_0 + (P_01 - P_11) * t_2)
  //            / ((V_01 - V_11) * t_0 + (V_11 - V_21) * t_2 + P_01 - P_21)
  //   3: t_2 = (((V_02 - V_12) * t_1 + P_22 - P_12) * t_0 + (P_02 - P_22) * t_1)
  //            / ((V_02 - V_22) * t_0 + (V_22 - V_12) * t_1 + P_02 - P_12)
  //   for t_0, t_1, t_2 not pairwise equal.
  // With some substitutions depending only on t_0 this gives
  //   1: 0 = (t_1 - t_0) * (f_2 - V_20 * t_2) - (t_2 - t_0) * (f_1 - V_10 * t_1)
  //   2: t_1 = (b_0 + b_1 * t_2) / (c_0 + c_1 * t_2)
  //   3: t_2 = (d_0 + d_1 * t_1) / (e_0 + e_1 * t_1)
  // And 3 in 2 gives:
  //   4: g_2 * t_1^2 - g_1 * t_1 - g_0 = 0
  // Then, with 4 and 3 in 1 and lengthy calculations with many substitutions (see constants k below, now independent of
  // t_0), the following polynomial can be constructed, with t_0 being a positive integer root of this polynomial.
  //   y = a_10 * x^10 + a_9 * x^9 + ... + a_0

  P_00 := Round(AHailstone1.Position.data[0]);
  P_01 := Round(AHailstone1.Position.data[1]);
  P_02 := Round(AHailstone1.Position.data[2]);
  P_10 := Round(AHailstone2.Position.data[0]);
  P_11 := Round(AHailstone2.Position.data[1]);
  P_12 := Round(AHailstone2.Position.data[2]);
  P_20 := Round(AHailstone3.Position.data[0]);
  P_21 := Round(AHailstone3.Position.data[1]);
  P_22 := Round(AHailstone3.Position.data[2]);
  V_00 := Round(AHailstone1.Velocity.data[0]);
  V_01 := Round(AHailstone1.Velocity.data[1]);
  V_02 := Round(AHailstone1.Velocity.data[2]);
  V_10 := Round(AHailstone2.Velocity.data[0]);
  V_11 := Round(AHailstone2.Velocity.data[1]);
  V_12 := Round(AHailstone2.Velocity.data[2]);
  V_20 := Round(AHailstone3.Velocity.data[0]);
  V_21 := Round(AHailstone3.Velocity.data[1]);
  V_22 := Round(AHailstone3.Velocity.data[2]);

  k[0] := P_00 - P_20;
  k[1] := P_00 - P_10;
  k[2] := P_11 - P_21;
  k[3] := P_01 - P_11;
  k[4] := P_01 - P_21;
  k[5] := P_22 - P_12;
  k[6] := P_02 - P_22;
  k[7] := P_02 - P_12;
  k[8] := V_11 - V_21;
  k[9] := V_22 - V_12;
  k[10] := V_01 - V_21;
  k[11] := V_01 - V_11;
  k[12] := V_02 - V_12;
  k[13] := V_02 - V_22;

  FH[0] := k[11] * k[9] + k[8] * k[12];
  FH[1] := k[4] * k[9] + k[8] * k[6];
  FH[2] := k[11] * k[13] - k[10] * k[12];
  FH[3] := k[11] * k[7] + k[4] * k[13] + k[8] * k[5] - k[2] * k[9] - k[10] * k[6] - k[3] * k[12];
  FH[4] := k[4] * k[7] - k[3] * k[6];
  FH[5] := k[10] * k[5] + k[2] * k[13];
  FH[6] := k[3] * k[5] + k[2] * k[7];

  k[14] := V_00 * k[9] - V_20 * k[12];
  k[15] := k[0] * k[9] - V_20 * k[6];
  k[16] := V_00 * k[13];
  k[17] := V_00 * k[7] + k[0] * k[13] - V_20 * k[5];
  k[18] := k[0] * k[7];
  k[19] := k[5] - k[7];
  k[20] := 2 * FH[2] * FH[3];
  k[21] := FH[3] * FH[3];
  k[22] := k[21] + 2 * FH[2] * FH[4];
  k[23] := 2 * FH[3] * FH[4];
  k[24] := 2 * FH[0] * FH[1];
  k[25] := FH[0] * FH[0];      // KILL?
  k[26] := FH[5] * k[25];    // KILL?
  k[126] := FH[5] * FH[0];
  k[127] := FH[5] * FH[1] + FH[6] * FH[0];
  k[128] := FH[6] * FH[1];
  k[27] := FH[5] * k[24] + FH[6] * k[25];   // KILL?
  k[28] := FH[1] * FH[1];                // KILL?
  k[29] := FH[5] * k[28] + FH[6] * k[24];   // KILL?
  k[30] := FH[6] * k[28];        // KILL?
  k[31] := FH[2] * FH[2];
  k[132] := k[20] + 4 * k[126];
  k[133] := k[22] + 4 * k[127];
  k[134] := k[23] + 4 * k[128];
  k[32] := k[31] + 4 * k[26];      // KILL?
  k[33] := k[20] + 4 * k[27];       // KILL?
  k[34] := k[22] + 4 * k[29];      // KILL?
  k[35] := k[23] + 4 * k[30];     // KILL?
  k[36] := k[31] + 2 * k[26];    // KILL?
  k[37] := k[20] + 2 * k[27];    // KILL?
  k[38] := k[22] + 2 * k[29];    // KILL?
  k[39] := k[23] + 2 * k[30];      // KILL?
  k[137] := k[20] + 2 * k[126];
  k[138] := k[22] + 2 * k[127];
  k[139] := k[23] + 2 * k[128];
  k[40] := k[14] + V_10 * (k[12] - k[9]);
  k[41] := k[15] + V_10 * k[6];
  k[42] := k[16] - k[14] - V_10 * k[13] - (k[12] - k[9]) * V_00;
  k[43] := k[17] - k[15] + V_10 * k[19] - (k[12] - k[9]) * k[1] - k[6] * V_00;
  k[44] := k[18] - k[6] * k[1];
  k[45] := k[42] * FH[0] - k[40] * FH[2];
  k[46] := k[42] * FH[1] + k[43] * FH[0] - k[41] * FH[2] - k[40] * FH[3];
  k[47] := k[43] * FH[1] + k[44] * FH[0] - k[41] * FH[3] - k[40] * FH[4];
  k[48] := k[44] * FH[1] - k[41] * FH[4];
  k[49] := k[42] * FH[2];
  k[50] := k[40] * k[31] - k[49] * FH[0];
  k[51] := k[42] * FH[3] + k[43] * FH[2];
  k[52] := k[40] * k[137] + k[41] * k[31] - k[51] * FH[0] - k[49] * FH[1];
  k[53] := k[42] * FH[4] + k[43] * FH[3] + k[44] * FH[2];
  k[54] := k[40] * k[138] + k[41] * k[137] - k[53] * FH[0] - k[51] * FH[1];
  k[55] := k[43] * FH[4] + k[44] * FH[3];
  k[56] := k[40] * k[139] + k[41] * k[138] - k[55] * FH[0] - k[53] * FH[1];
  k[57] := k[44] * FH[4];
  k[58] := FH[4] * FH[4];
  k[59] := k[40] * k[58] + k[41] * k[139] - k[57] * FH[0] - k[55] * FH[1];
  k[60] := k[41] * k[58] - k[57] * FH[1];
  k[61] := k[13] * V_00 - k[16];
  k[62] := 2 * k[25] * k[61];
  k[63] := k[13] * k[1] - k[19] * V_00 - k[17];
  k[64] := 2 * (k[24] * k[61] + k[25] * k[63]);
  k[65] := - k[19] * k[1] - k[18];
  k[66] := 2 * (k[28] * k[61] + k[24] * k[63] + k[25] * k[65]);
  k[67] := 2 * (k[28] * k[63] + k[24] * k[65]);
  k[68] := 2 * k[28] * k[65];
  k[69] := k[50] + k[62];
  k[70] := k[52] + k[64];
  k[71] := k[54] + k[66];
  k[72] := k[56] + k[67];
  k[73] := k[59] + k[68];
  k[74] := k[45] * k[45];
  k[75] := 2 * k[45] * k[46];
  k[76] := k[46] * k[46] + 2 * k[45] * k[47];
  k[77] := 2 * (k[45] * k[48] + k[46] * k[47]);
  k[78] := k[47] * k[47] + 2 * k[46] * k[48];
  k[79] := 2 * k[47] * k[48];
  k[80] := k[48] * k[48];

  FA[0] := k[58] * k[80] - k[60] * k[60];
  FA[1] := k[134] * k[80] + k[58] * k[79] - 2 * k[73] * k[60];
  FA[2] := k[133] * k[80] + k[134] * k[79] + k[58] * k[78] - k[73] * k[73] - 2 * k[72] * k[60];
  FA[3] := k[133] * k[79] + k[134] * k[78] + k[58] * k[77] + k[132] * k[80]
    - 2 * (k[71] * k[60] + k[72] * k[73]);
  FA[4] := k[31] * k[80] + k[133] * k[78] + k[134] * k[77] + k[58] * k[76] + k[132] * k[79] - k[72] * k[72]
    - 2 * (k[70] * k[60] + k[71] * k[73]);
  FA[5] := k[31] * k[79] + k[133] * k[77] + k[134] * k[76] + k[58] * k[75] + k[132] * k[78]
    - 2 * (k[69] * k[60] + k[70] * k[73] + k[71] * k[72]);
  FA[6] := k[31] * k[78] + k[133] * k[76] + k[134] * k[75] + k[58] * k[74] + k[132] * k[77] - k[71] * k[71]
    - 2 * (k[69] * k[73] + k[70] * k[72]);
  FA[7] := k[31] * k[77] + k[133] * k[75] + k[134] * k[74] + k[132] * k[76] - 2 * (k[69] * k[72] + k[70] * k[71]);
  FA[8] := k[31] * k[76] + k[132] * k[75] + k[133] * k[74] - k[70] * k[70] - 2 * k[69] * k[71];
  FA[9] := k[31] * k[75] + k[132] * k[74] - 2 * k[69] * k[70];
  FA[10] := k[31] * k[74] - k[69] * k[69];

  // Debug calculations
  //a_1 := V_00 * t_0 + P_00 - P_20;
  //a_2 := V_00 * t_0 + P_00 - P_10;
  //b_0 := (P_11 - P_21) * t_0;
  //b_1 := (V_01 - V_21) * t_0 + P_01 - P_11;
  //c_0 := (V_01 - V_11) * t_0 + P_01 - P_21;
  //c_1 := V_11 - V_21;
  //d_0 := (P_22 - P_12) * t_0;
  //d_1 := (V_02 - V_12) * t_0 + P_02 - P_22;
  //e_0 := (V_02 - V_22) * t_0 + P_02 - P_12;
  //e_1 := V_22 - V_12;
  //f_2 := c_0 * e_1 + c_1 * d_1;
  //f_1 := c_0 * e_0 + c_1 * d_0 - b_0 * e_1 - b_1 * d_1;
  //f_0 := b_1 * d_0 + b_0 * e_0;
  //
  //act := f_2 * t_1 * t_1 + f_1 * t_1 - f_0;
  //Write('debug10: ', 0 = act, '    ');
  //
  //if f_2 <> 0 then
  //begin
  //  act := Round(- f_1 / (2 * f_2) + Sqrt((f_1 / (2 * f_2)) * (f_1 / (2 * f_2)) + f_0 / f_2));
  //  Write('debug15: ', t_1 = act);
  //  act := Round(- f_1 / (2 * f_2) - Sqrt((f_1 / (2 * f_2)) * (f_1 / (2 * f_2)) + f_0 / f_2));
  //  Write(' OR ', t_1 = act, '    ');
  //end;
  //
  //act := (e_0 + e_1 * t_1) * t_2 - (d_0 + d_1 * t_1);
  //Write('debug20: ', 0 = act, '    ');
  //
  //act := (a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * t_1 * t_1
  //  + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1  - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * t_1
  //  + t_0 * (V_20 * d_0 - a_1 * e_0) + (e_0 * t_0 - d_0) * a_2;
  //Write('debug30: ', 0 = act, '    ');
  //
  //act := Round((a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * (f_1 * f_1 + 2 * f_0 * f_2 - f_1 * Sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //  + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * (- f_1 * f_2 + f_2 * Sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //  + t_0 * (V_20 * d_0 - a_1 * e_0) * 2 * f_2 * f_2 + (e_0 * t_0 - d_0) * a_2 * 2 * f_2 * f_2);
  //Write('debug40: ', 0 = act, '    ');
  //
  //Write('debug41: ',
  //  a_1 * k[9] - V_20 * d_1
  //  = k[14] * t_0 + k[15], '    ');
  //Write('debug42: ',
  //  d_1 - k[9] * t_0
  //  = (k[12] - k[9]) * t_0 + k[6], '    ');
  //Write('debug43: ',
  //  a_1 * e_0 - V_20 * d_0
  //  = k[16] * t_0 * t_0 + k[17] * t_0 + k[18], '    ');
  //Write('debug44: ',
  //  d_0 - e_0 * t_0
  //  = - k[13] * t_0 * t_0 + k[19] * t_0, '    ');
  //Write('debug45: ',
  //  f_1 * f_1
  //  = FH[2] * FH[2] * t_0 * t_0 * t_0 * t_0 + k[20] * t_0 * t_0 * t_0 + k[22] * t_0 * t_0 + k[23] * t_0 + FH[4] * FH[4], '    ');
  //Write('debug46: ',
  //  f_2 * f_2
  //  = FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1], '    ');
  //Write('debug47: ',
  //  f_0 * f_2
  //  = k[126] * t_0 * t_0 * t_0 + k[127] * t_0 * t_0 + k[128] * t_0, '    ');
  //Write('debug48: ',
  //  f_1 * f_1 + 4 * f_0 * f_2
  //  = k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58], '    ');
  //Write('debug49: ',
  //  f_1 * f_1 + 2 * f_0 * f_2
  //  = k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58], '    ');
  //
  //act := Round((k[14] * t_0 + k[15] + V_10 * ((k[12] - k[9]) * t_0 + k[6])) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58] - f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //  + (k[16] * t_0 * t_0 + k[17] * t_0 + k[18] - t_0 * (k[14] * t_0 + k[15]) - ((k[12] - k[9]) * t_0 + k[6]) * a_2 - V_10 * (k[13] * t_0 * t_0 - k[19] * t_0)) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //  - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]));
  //Write('debug50: ', 0 = act, '    ');
  //
  //Write('debug53: ',
  //  0 = Round((k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58] - f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //    + ((k[16] - k[14] - V_10 * k[13] - (k[12] - k[9]) * V_00) * t_0 * t_0 + (k[17] - k[15] + V_10 * k[19] - (k[12] - k[9]) * k[1] - k[6] * V_00) * t_0 + k[18] - k[6] * k[1]) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //    - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1])),
  //    '    ');
  //
  //Write('debug55: ',
  //  0 = Round((k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58])
  //    - (k[40] * t_0 + k[41]) * f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2)
  //    + (k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //    - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1])),
  //    '    ');
  //
  //Write('debug70: ',
  //  0 = Round(((k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (FH[0] * t_0 + FH[1]) - (k[40] * t_0 + k[41]) * (FH[2] * t_0 * t_0 + FH[3] * t_0 + FH[4])) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
  //    + (k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58])
  //    - (k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (FH[2] * t_0 * t_0 + FH[3] * t_0 + FH[4]) * (FH[0] * t_0 + FH[1])
  //    - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * (V_00 * t_0 + k[1]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]),
  //    '    ');
//
//  Write('debug73: ',
//    0 = Round((
//        (k[42] * FH[0] - k[40] * FH[2]) * t_0 * t_0 * t_0
//        + (k[42] * FH[1] + k[43] * FH[0] - k[41] * FH[2] - k[40] * FH[3]) * t_0 * t_0
//        + (k[43] * FH[1] + k[44] * FH[0] - k[41] * FH[3] - k[40] * FH[4]) * t_0
//        + k[44] * FH[1] - k[41] * FH[4]
//    ) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
//    + (k[40] * k[31] - k[42] * FH[2] * FH[0]) * t_0 * t_0 * t_0 * t_0 * t_0
//    + (k[40] * k[137] + k[41] * k[31] - k[42] * FH[3] * FH[0] - k[43] * FH[2] * FH[0] - k[42] * FH[2] * FH[1]) * t_0 * t_0 * t_0 * t_0
//    + (k[40] * k[138] + k[41] * k[137] - k[42] * FH[4] * FH[0] - k[43] * FH[3] * FH[0] - k[44] * FH[2] * FH[0] - k[42] * FH[3] * FH[1] - k[43] * FH[2] * FH[1]) * t_0 * t_0 * t_0
//    + (k[40] * k[139] + k[41] * k[138] - k[43] * FH[4] * FH[0] - k[44] * FH[3] * FH[0] - k[42] * FH[4] * FH[1] - k[43] * FH[3] * FH[1] - k[44] * FH[2] * FH[1]) * t_0 * t_0
//    + (k[40] * k[58] + k[41] * k[139] - k[44] * FH[4] * FH[0] - k[43] * FH[4] * FH[1] - k[44] * FH[3] * FH[1]) * t_0
//    + k[41] * k[58] - k[44] * FH[4] * FH[1]
//    + 2 * (k[13] * V_00 * FH[0] * FH[0] - k[16] * FH[0] * FH[0]) * t_0 * t_0 * t_0 * t_0 * t_0
//    + 2 * (k[13] * V_00 * k[24] + k[13] * k[1] * FH[0] * FH[0] - k[19] * V_00 * FH[0] * FH[0] - k[16] * k[24] - k[17] * FH[0] * FH[0]) * t_0 * t_0 * t_0 * t_0
//    + 2 * (k[13] * V_00 * FH[1] * FH[1] + k[13] * k[1] * k[24] - k[19] * V_00 * k[24] - k[19] * k[1] * FH[0] * FH[0] - k[16] * FH[1] * FH[1] - k[17] * k[24] - k[18] * FH[0] * FH[0]) * t_0 * t_0 * t_0
//    + 2 * (k[13] * k[1] * FH[1] * FH[1] - k[19] * V_00 * FH[1] * FH[1] - k[19] * k[1] * k[24] - k[17] * FH[1] * FH[1] - k[18] * k[24]) * t_0 * t_0
//    + 2 * (- k[19] * k[1] * FH[1] * FH[1] - k[18] * FH[1] * FH[1]) * t_0,
//    '    ');
//
//  Write('debug78: ',
//    0 = Round((k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
//      + (k[50] + k[62]) * t_0 * t_0 * t_0 * t_0 * t_0 + (k[52] + k[64]) * t_0 * t_0 * t_0 * t_0 + (k[54] + k[66]) * t_0 * t_0 * t_0 + (k[56] + k[67]) * t_0 * t_0 + (k[59] + k[68]) * t_0 + k[60],
//      '    ');
//
//  Write('debug80: ',
//    0 = Round((k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * sqrt(k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58])
//      + k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]),
//      '    ');
//  WriteLn;
//  WriteLn('  0 = ((', k[45], ') * x^3 + (', k[46], ') * x^2 + (', k[47], ') * x + (', k[48], ')) * sqrt((', k[31], ') * x^4 + (', k[132], ') * x^3 + (', k[133], ') * x^2 + (', k[134], ') * x + (', k[58], ')) + (',
//    k[69], ') * x^5 + (', k[70], ') * x^4 + (', k[71], ') * x^3 + (', k[72], ') * x^2 + (', k[73], ') * x + (', k[60], ')');

  Write('debug83: ',
    (k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * (k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58]) =
      (k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]) * (k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]),
      '    ');
  Write('debug85: ',
    0 =
      (
          k[45] * k[45] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
          + 2 * k[45] * k[46] * t_0 * t_0 * t_0 * t_0 * t_0
          + k[46] * k[46] * t_0 * t_0 * t_0 * t_0
          + 2 * k[45] * k[47] * t_0 * t_0 * t_0 * t_0
          + 2 * k[45] * k[48] * t_0 * t_0 * t_0
          + 2 * k[46] * k[47] * t_0 * t_0 * t_0
          + k[47] * k[47] * t_0 * t_0
          + 2 * k[46] * k[48] * t_0 * t_0
          + 2 * k[47] * k[48] * t_0
          + k[48] * k[48]
      ) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58])
      - k[69] * k[69] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
      - 2 * k[69] * k[70] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
      - (k[70] * k[70] + 2 * k[69] * k[71]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
      - 2 * (k[69] * k[72] + k[70] * k[71]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
      - (k[71] * k[71] + 2 * k[69] * k[73] + 2 * k[70] * k[72]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
      - 2 * (k[69] * k[60] + k[70] * k[73] + k[71] * k[72]) * t_0 * t_0 * t_0 * t_0 * t_0
      - (k[72] * k[72] + 2 * k[70] * k[60] + 2 * k[71] * k[73]) * t_0 * t_0 * t_0 * t_0
      - 2 * (k[71] * k[60] + k[72] * k[73]) * t_0 * t_0 * t_0
      - (k[73] * k[73] + 2 * k[72] * k[60]) * t_0 * t_0
      - 2 * k[73] * k[60] * t_0
      - k[60] * k[60],
      '    ');

  WriteLn('debug96: ', EvaluateAt(t_0) = 0);

  NormalizeCoefficients;

  WriteLn('debug99: ', EvaluateAt(t_0) = 0, '    ');
end;

function TFirstCollisionPolynomial.EvaluateAt(const AT0: Int64): TBigInt;
var
  i: Low(FA)..High(FA);
begin
  Result := TBigInt.Zero;
  for i := High(FA) downto Low(FA) do
    Result := Result * AT0 + FA[i];
end;

function TFirstCollisionPolynomial.CalcPositiveIntegerRoot: Int64;
var
  dividers: TDividers;
  factors: TInt64Array;
  divider: Int64;
begin
  Result := 0;
  //factors := TIntegerFactorization.PollardsRhoAlgorithm(FA[0]);
  //dividers := TDividers.Create(factors);
  //
  //try
  //for divider in dividers do
  //begin
  //  //WriteLn('Check if ', divider, ' is a root...');
  //  if EvaluateAt(divider) = 0 then
  //  begin
  //    Result := divider;
  //    Break;
  //  end;
  //end;
  //
  //finally
  //  dividers.Free;
  //end;
end;

function TFirstCollisionPolynomial.CalcT1(const AT0: Int64): Int64;
var
  g_0, g_1, g_2: Int64;
  g: Extended;
begin
  //g_2 := FH[0] * AT0 + FH[1];
  //g_1 := FH[2] * AT0 * AT0 + FH[3] * AT0 + FH[4];
  //g_0 := FH[5] * AT0 * AT0 + FH[6] * AT0;
  //g := - g_1 / (2 * g_2);
  //Result := Round(g + sqrt(g * g + g_0));
end;

{ TNeverTellMeTheOdds }

function TNeverTellMeTheOdds.AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
var
  m1, m2, x, y: Double;
begin
  Result := False;
  m1 := AHailstone1.Velocity.data[1] / AHailstone1.Velocity.data[0];
  m2 := AHailstone2.Velocity.data[1] / AHailstone2.Velocity.data[0];
  if m1 <> m2 then
  begin
    x := (AHailstone2.Position.data[1] - m2 * AHailstone2.Position.data[0]
      - AHailstone1.Position.data[1] + m1 * AHailstone1.Position.data[0])
      / (m1 - m2);
    if (FMin <= x) and (x <= FMax)
    and (x * Sign(AHailstone1.Velocity.data[0]) >= AHailstone1.Position.data[0] * Sign(AHailstone1.Velocity.data[0]))
    and (x * Sign(AHailstone2.Velocity.data[0]) >= AHailstone2.Position.data[0] * Sign(AHailstone2.Velocity.data[0]))
    then
    begin
      y := m1 * (x - AHailstone1.Position.data[0]) + AHailstone1.Position.data[1];
      if (FMin <= y) and (y <= FMax) then
        Result := True
    end;
  end;
end;

// For debug calculations:
Const
  T : array[0..4] of Byte = (5, 3, 4, 6, 1);

procedure TNeverTellMeTheOdds.FindRockThrow(const AIndex1, AIndex2, AIndex3: Integer);
var
  //i, j, k: Integer;
  //x0, x1, x2: Extended;
  f: TFirstCollisionPolynomial;
  t0, t1: Int64;
  p, v: Tvector3_extended;
  test: TBigInt;
begin
  WriteLn;
  WriteLn(AIndex1, ' ', AIndex2, ' ', AIndex3);
  f := TFirstCollisionPolynomial.Create;
  f.Init(FHailstones[AIndex1], FHailstones[AIndex2], FHailstones[AIndex3], T[AIndex1], T[AIndex2], T[AIndex3]);
  //t0 := f.CalcPositiveIntegerRoot;
  //WriteLn('t0: ', t0, ' ', t0 = T[AIndex1]);
  //t1 := f.CalcT1(t0);
  //WriteLn(', t1: ', t1);
  f.Free;

  //// V_x = (V_0 * t_0 - V_1 * t_1 + P_0 - P_1) / (t_0 - t_1)
  //v := (FHailstones[AIndex1].Velocity * t0 - FHailstones[AIndex2].Velocity * t1
  //  + FHailstones[AIndex1].Position - FHailstones[AIndex2].Position) / (t0 - t1);
  //// P_x = (V_0 - V_x) * t_0 + P_0
  //p := (FHailstones[AIndex1].Velocity - v) * t0 + FHailstones[AIndex1].Position;
  //FPart2 := Round(p.data[0]) + Round(p.data[1]) + Round(p.data[2]);

  //for i := 0 to FHailstones.Count - 3 do
  //  for j := i + 1 to FHailstones.Count - 2 do
  //    for k:= j + 1 to FHailstones.Count - 1 do
  //    begin
  //      WriteLn(i, j, k);
  //      solver := TRockThrowSolver.Create(FHailstones[i], FHailstones[j], FHailstones[k], 0);
  //      case i of
  //        0: x0 := 5;
  //        1: x0 := 3;
  //        2: x0 := 4;
  //      end;
  //      f := solver.CalcValue(x0);
  //      solver.Free;
  //    end;

  //for i := 80 to 120 do
  //begin
  //  solver := TRockThrowSolver.Create(FHailstones[0], FHailstones[1], FHailstones[2], 0);
  //  x0 := i / 20;
  //  f := solver.CalcValue(x0);
  //  WriteLn(x0, ' ', f.Valid, ' ', f.Value);
  //  solver.Free;
  //end;
end;

constructor TNeverTellMeTheOdds.Create(const AMin: Int64; const AMax: Int64);
begin
  FMin := AMin;
  FMax := AMax;
  FHailstones := THailstones.Create;
end;

destructor TNeverTellMeTheOdds.Destroy;
begin
  FHailstones.Free;
  inherited Destroy;
end;

procedure TNeverTellMeTheOdds.ProcessDataLine(const ALine: string);
begin
  FHailstones.Add(THailstone.Create(ALine));
end;

procedure TNeverTellMeTheOdds.Finish;
var
  i, j, k: Integer;
begin
  for i := 0 to FHailstones.Count - 2 do
    for j := i + 1 to FHailstones.Count - 1 do
      if AreIntersecting(FHailstones[i], FHailstones[j]) then
        Inc(FPart1);

  for i := 0 to FHailstones.Count - 1 do
    for j := 0 to FHailstones.Count - 1 do
      for k := 0 to FHailstones.Count - 1 do
        if (i <> j) and (i <> k) and (j <> k) then
          FindRockThrow(i, j, k);
  //FindRockThrow(0, 1, 2);
end;

function TNeverTellMeTheOdds.GetDataFileName: string;
begin
  Result := 'never_tell_me_the_odds.txt';
end;

function TNeverTellMeTheOdds.GetPuzzleName: string;
begin
  Result := 'Day 24: Never Tell Me The Odds';
end;

end.