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Optimal Pyramids Play


by testmatch

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     I've immersed myself in the wonder of the Pyramids, been visited by the queens and kings of the Lost Desert and started dreaming in cards. That's right, I've been maddened by the variant of Solitaire that a certain Aisha has been peddling in front of the Lost Desert's biggest landmark, Pyramids!

     Fear not, with my trustworthy N-4 Info Retrieval Bot, JesN4, and a handy issue of the Neopian Times from year 8, I've compiled a new set of tips. I hope I can save you from the dreadful dreams that I shared with last year's reporter on the Pyramids.

     First, realise that you are going to lose a lot. Using my four simple tips, you'll only win 3.28% of games. That is roughly 2.5 times harder than Sakhmet Solitaire (tip use 3 cards per draw). And yes, you heard me right, you only need four tips. Skip ahead if numbers scares you.

     Likelihood of winning

     Fidgeting around with JesN4, I realised I could get it to play some games in its head. So, being a good Petpet owner, I told it to play with a couple different strategies for a few million games. Don't worry, JesN4 only spent a couple hours playing before it went back to retrieving information.

     You're gonna play a lot of games to get the Pyramids bonus gold trophy. That's two wins in a row, which happens once every 929 games on average (opposed to 132 games with Sakhmet Solitaire's bonus gold trophy).

     Although you should win 3 games in 100, just based on luck, 36 out of 100 Neopians will not. In fact, 4 out of 100 Neopians won't win any games at all after one hundred games. That's assuming you're playing without any mistakes!

     Just to allay any fears you have while playing, here's the chances you won't even get to 4 wins after playing too many games.

     

     Games Played

     Probability of 3 or fewer wins

     

     50

     91.88%

      100

     58.37%

      150

     27.18%

      200

     10.39%

      250

     3.49%

     

     And if you're going for silver (that's five wins in total), here's some data to calm your nerves.

     

     Games Played

     Probability of less than 5 wins

     

     100

     76.84%

      125

     60.90%

      150

     45.19%

      200

     21.24%

      250

     8.52%

      500

     0.03%

     

     Yes, some unlucky Neopians are going be sunbaked before they get this trophy.

     If you want more information on the chances of winning, I suggest finding your own JesN4, or reading up on Mathematics in Space (particularly the bit on combinations in the probability chapter). Anyway, here are the four tips.

     Tip 1: If you have to choose, go with the card at the bottom.

      x

      x x

      x x x

      9 2 x x

      _ _ _ x x

     _ _ _ _ J 5

     Choosing between the nine and the Jack, go for the Jack here.

      Tip 2: If you've got a tie, go with the card that will reveal more cards above it.

      x

      x x

      x x x

      x x x x

      x x x x x

     A 7 A _ 9 3

     Choosing between 7 and 9, go for the nine here.

      Tip 3: If you've still got a tie, choose the one closer to the center of the row.

      x

      x x

      x x x

      x x x x

      x x x x x

     A 7 4 Q 9 3

     Choosing between Ace and Queen, go for the Queen here.

      Tip 4: If you've still (still) got a tie, just choose whichever you fancy.

     That is it. Don't worry about consecutive cards, they'll obviously be removed in your next turn. Don't worry about anything else, the chances it'll matter are too low. Read on if you're curious, if not, play some Pyramids, enjoy the light and pack lots of sunscreen.

     The Logic

     Tip 1 is pretty obvious intuition. If you don't get rid of cards at the bottom, they'll leave a trail of unknown cards that'll haunt you in the future.

     Tip 2 is also pretty simple intuition. This will give you more cards to pick, and you need to remove as many cards as possible while you still got cards in the pile.

     Tip 3 is also pretty simple, the ones closer to the center are more likely to lead to more cards being removed, by way of tip 2. That is, since the edge pieces of the pyramid can only reveal one card above them, they are given lower priority when choosing the card to remove. After all, the card they reveal will just be another edge piece.

     Tip 4 is kinda mind blowing. And honestly, the only way to improve upon this strategy would be to card count (i.e. keep track of all the cards you remove) and then do complex math to see which is more likely to produce a winning result. Not only is this unrealistic, but it probably won't improve your wins by much.

     Now, you may be thinking, how about going for the consecutive cards? Good point, I tried to get JesN4 to consider that, but unfortunately it doesn't seem to help. After playing 10,000,001 games using the normal strategy, and replaying whilst going for consecutive cards, I found that considering consecutive cards would decrease your chances of winning.

     I'm in as much puzzlement as the next Neopian. I think the intuition is that consecutive cards will have a greater chance of being removed. For example, imagine the only visible cards were 4, 5, 6, and 8. With 8, only 7 and 9 can help remove it. With 4, 5, 6, picking up a 3, 4, 5, 6, or 7 will help remove them. Thus, its more important to pick the 8, while you still have cards to remove it. The consecutive cards will always be picked up consecutively using this technique anyway.

      x

      x x

      x x x

      5 4 8 6

      If you really want to continue considering consecutive cards, I won't stop you. You'll win more Neopoints from the game, and won't lose many more games. JesN4 only lost 150 more games after 10,000,001 games, that's about a 0.04% change in wins (or 0.0015% more losses). That is tiny!

     Be warned though, whatever the reason for the lower win rate, it is tied to the game mechanics. Yes, I tried prioritising the value of consecutive cards to varying amounts. It didn't help. Looking at how JesN4 played, the 150 losses were accounted for by over 150 games. That means that if you considered consecutive cards, you would win some games that my strategy would not win; however, my strategy just won more games overall. It seems there are few games where choosing consecutive cards first actually improves your chances of winning.

     I could, of course, be wrong. There are 52! possible games or 8x1067, so getting JesN4 to play millions of games may not do much (that's a lot less than 1% of all possible games). JesN4 did play random games though, and while 10 million is a tiny amount of potential games, its still more than you're going to play. Also any stronger strategy will be way more complicated than mine, which isn't worth it. Maybe if it gets you a 20% win rate, but you're gonna burn some paper.

     Oh my, I hope you aren't tempted to find a better strategy. If you seek to find more secrets about the Pyramids, a few words to the wise. I have tried getting JesN4 to do some card counting, then check to see if the game becomes impossible if you don't choose a card. It didn't change results at all. Not after 10,000,001 games. Maybe it might matter, but the games where it matters are rare. Rarer than 10,000,001 games. Also statistics on Sakhmet Solitaire were determined experimentally, expect the same with Pyramids.

     With that, I bid you farewell, and may your dreams be peaceful.

     

 
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