Ice Cold Decks

5 Oct 2009

Any idiot can stack a cold deck to trick an opponent in Texas Hold'em Poker into losing all his chips. However usually the player on the right of the dealer cuts the deck, which means the cheater must swap the deck after this point, which makes it much harder to do subtly.

I do not condone cheating in poker, it is immoral, dangerous and often illegal, but out of curiosity I was wondering if it is possible to stack a deck so that no matter which way it is cut, the winning hand is always dealt to the dealer. This would be fun to use as a magic trick, to be able to consistently predict the winning hand. I was skeptical at first, but after running a hastily written script for about half an hour I found such a deck for 2 player heads up:

  1. A spades
  2. T spades
  3. 4 clubs
  4. 2 clubs
  5. K clubs
  6. 9 hearts
  7. 8 hearts
  8. 9 spades
  9. 8 clubs
  10. T clubs
  11. 8 diamonds
  12. 9 clubs
  13. 8 spades
  14. 9 diamonds
  15. 4 diamonds
  16. Q diamonds
  17. 5 diamonds
  18. T hearts
  19. 5 clubs
  20. T diamonds
  21. 7 hearts
  22. A diamonds
  23. 6 diamonds
  24. A clubs
  25. 4 spades
  26. J hearts
  27. K spades
  28. A hearts
  29. 3 hearts
  30. K diamonds
  31. 7 diamonds
  32. 6 hearts
  33. 5 hearts
  34. K hearts
  35. 3 diamonds
  36. Q hearts
  37. 3 clubs
  38. 2 hearts
  39. Q spades
  40. J clubs
  41. 7 spades
  42. J diamonds
  43. 3 spades
  44. 6 spades
  45. 6 clubs
  46. 7 clubs
  47. 4 hearts
  48. 2 spades
  49. 5 spades
  50. J spades
  51. Q clubs
  52. 2 diamonds

(hover your mouse over the cards to see the hands that would be dealt)

Give it a go: stack a deck like this (so the 10 of diamonds is on the top of the deck), cut it in any way, deal cards to two players in the usual fashion, deal the community cards (don't forget the burned cards), and you will find that the dealer always wins.

After this, I began wondering, is it possible to do this with 3 players? A few speed improvements to the script and I got just that:

  1. A spades
  2. 8 clubs
  3. T hearts
  4. A diamonds
  5. 4 diamonds
  6. K diamonds
  7. 2 hearts
  8. 4 clubs
  9. Q clubs
  10. 9 diamonds
  11. 7 clubs
  12. 5 clubs
  13. 2 clubs
  14. J hearts
  15. Q hearts
  16. 2 spades
  17. J clubs
  18. 8 spades
  19. 9 clubs
  20. K hearts
  21. 9 hearts
  22. A hearts
  23. T spades
  24. 3 hearts
  25. 9 spades
  26. 3 spades
  27. 7 spades
  28. 6 clubs
  29. 6 spades
  30. 7 hearts
  31. 8 diamonds
  32. 4 hearts
  33. 6 diamonds
  34. T diamonds
  35. 2 diamonds
  36. J spades
  37. 3 clubs
  38. 5 hearts
  39. 7 diamonds
  40. 3 diamonds
  41. A clubs
  42. T clubs
  43. Q diamonds
  44. 4 spades
  45. 5 diamonds
  46. 6 hearts
  47. K clubs
  48. 8 hearts
  49. Q spades
  50. 5 spades
  51. J diamonds
  52. K spades

And 4 players, though this one took the computer about 6 hours to find:

  1. A spades
  2. 3 spades
  3. 4 hearts
  4. 9 hearts
  5. 7 hearts
  6. 8 spades
  7. 6 hearts
  8. 5 spades
  9. 6 spades
  10. J hearts
  11. 5 diamonds
  12. T spades
  13. Q diamonds
  14. K diamonds
  15. K spades
  16. 2 diamonds
  17. 7 diamonds
  18. T diamonds
  19. 4 diamonds
  20. 8 clubs
  21. 6 diamonds
  22. 9 clubs
  23. 3 diamonds
  24. 3 clubs
  25. 5 clubs
  26. A diamonds
  27. 2 spades
  28. 2 clubs
  29. 4 spades
  30. 6 clubs
  31. 5 hearts
  32. Q clubs
  33. Q hearts
  34. K clubs
  35. T hearts
  36. 9 diamonds
  37. 3 hearts
  38. Q spades
  39. 8 hearts
  40. K hearts
  41. A hearts
  42. 9 spades
  43. J spades
  44. 4 clubs
  45. 7 spades
  46. J clubs
  47. A clubs
  48. 8 diamonds
  49. 2 hearts
  50. T clubs
  51. J diamonds
  52. 7 clubs

I've tried searching for a 5 player deck, but at the moment the heuristic has only found one where the dealer wins 50/52 of the possible cuts...

Five player deck now solved!

  1. A spades
  2. 6 clubs
  3. 4 spades
  4. 8 hearts
  5. 2 spades
  6. T hearts
  7. 3 diamonds
  8. Q hearts
  9. 7 diamonds
  10. K hearts
  11. 9 diamonds
  12. 5 hearts
  13. J diamonds
  14. A hearts
  15. 6 diamonds
  16. 4 hearts
  17. 8 clubs
  18. 2 hearts
  19. T clubs
  20. 3 spades
  21. Q clubs
  22. 7 spades
  23. K clubs
  24. 9 spades
  25. 5 clubs
  26. J spades
  27. A clubs
  28. 6 spades
  29. 4 clubs
  30. 8 diamonds
  31. 2 clubs
  32. T diamonds
  33. 3 hearts
  34. Q diamonds
  35. 7 hearts
  36. K diamonds
  37. 9 hearts
  38. 5 diamonds
  39. J hearts
  40. A diamonds
  41. 6 hearts
  42. 4 diamonds
  43. 8 spades
  44. 2 diamonds
  45. T spades
  46. 3 clubs
  47. Q spades
  48. 7 clubs
  49. K spades
  50. 9 clubs
  51. 5 spades
  52. J clubs

Update 19/2/10 - A special case constructed by logic (not with a computer) has been devised for 6 player small blind by a man named Tom Darrow [link], can similar techniques be used to solve other decks?

  1. A spades
  2. Q diamonds
  3. T spades
  4. 8 spades
  5. 6 diamonds
  6. 4 spades
  7. 2 spades
  8. K spades
  9. J hearts
  10. 9 hearts
  11. 7 hearts
  12. 5 hearts
  13. 3 hearts
  14. A clubs
  15. Q clubs
  16. T hearts
  17. 8 clubs
  18. 6 clubs
  19. 4 clubs
  20. 2 clubs
  21. K diamonds
  22. J diamonds
  23. 9 diamonds
  24. 7 diamonds
  25. 5 diamonds
  26. 3 diamonds
  27. A hearts
  28. Q spades
  29. T clubs
  30. 8 hearts
  31. 6 spades
  32. 4 hearts
  33. 2 hearts
  34. K clubs
  35. J spades
  36. 9 spades
  37. 7 spades
  38. 5 spades
  39. 3 spades
  40. A diamonds
  41. Q hearts
  42. T diamonds
  43. 8 diamonds
  44. 6 hearts
  45. 4 diamonds
  46. 2 diamonds
  47. K hearts
  48. J clubs
  49. 9 clubs
  50. 7 clubs
  51. 5 clubs
  52. 3 clubs

It turns out some really high number of players (11, 12, 16, 18) have easily solvable solutions for particular players, the question now remains: how much of the table below can be made green?

Click the table cells see the decks.

2 52 52
3 52 52 52
4 52 52 52 52
5 52 52 52 52
6 52 52
7
8
9 52
10 52 52
11 52 52 52
12 52
13
14 52
15 52
16 52
17
18 52
19
20 52
21 52
22

This magic trick has been featured on Scam School.