Mahjong Solitaire Strategy Guide

This guide is written for experienced Mahjong Solitaire players seeking to enhance their playing skills and deepen their understanding of the game. If you are a beginner, please read our introductory text on how to play Mahjong Solitaire and have some practice.

This article focuses on the traditional Turtle layout, also known as the Spider layout (see Figure 1). Other layouts have different difficulty levels and may require different strategies. However, the advice provided in this article should help you understand the game in general and, as a result, improve your play for other layouts as well.

Various games employ different strategies to create deals (i.e., shuffling and arranging the tiles in a particular order). Here, we will focus on the classic version of Mahjong Solitaire in which tiles in every deal are arranged completely randomly.

Is Mahjong Solitaire difficult?

To know how hard Mahjong Solitaire is, we need to measure its difficulty. Let’s, for example, try playing the game by choosing matching tiles randomly. Don’t use any strategies. In fact, do not think at all – just pick the first pair you see. How often can you win when you play this way? If your win ratio is high, then the game is easy because it doesn't require much thinking or planning. But if the ratio is low, more advanced strategies are needed to consistently win, which means the game is harder. This method of measuring difficulty is not perfect, but it serves as a good starting point.

So, is Mahjong Solitaire difficult? Yes, it is. If you select tiles randomly, as described above, you have only around a 5.9% chance of winning the game. (This is true only for the deals in which tiles are randomly shuffled. If the game is set to a specific difficulty level or to make deals solvable, then this percentage may be different. Also, the 5.9% applies only to the Turtle layout, and for other layouts, this number can be higher or lower.)

The difficulty varies tremendously from deal to deal. If you replay the same deal multiple times, you can calculate the deal’s win ratio. It turns out that some deals are very easy – they can have even an 80% chance of winning when playing purely randomly. This, however, happens rarely. Figure 2 shows how often various deal difficulties occur.

Is there a way to improve the odds of winning? You can use popular strategies often mentioned on the Internet, such as reducing the length of the longest rows, maximizing the number of available moves, and avoiding potentially blocking situations. However, this does not get you far: in practice, you can increase your overall win ratio twofold, maybe threefold, and reach the neighborhood of 20%. If you add undos into the mix, backtrack and play different moves after learning about the tiles hidden in the deeper layers, you may improve your chances even further. But you will never reach 100%.

Is Mahjong Solitaire always winnable?

Mahjong Solitaire is not always winnable. There are situations when three tiles of the same kind are stacked on top of each other, and there is no way to remove them. How often does it happen? Or, more generally, how many deals can be won and how many can’t, no matter what the player does?

In addition to the obviously unsolvable deals, such as the one with the three stacked tiles, there are many deals with much more complicated blocking patterns. Therefore, it is sometimes not easy to determine if a deal can’t be solved. For such a deal, we can only say that it has not been solved, despite our best efforts to solve it.

And how many deals can be solved? The most recent estimate is at least 96%. If your success ratio is approaching this number, you must be a very good player. And you should not worry if you can’t exceed it – because if you do, there is something suspect going on. Maybe the deals are not random? Or perhaps you shuffle the tiles when desperate?

Two sure strategies

Let’s say you have a solution. You found a sequence of pair removals that clears the board. Is it possible to break this solution by removing the same pairs but in a different order? No, the order in which you remove the pairs does not matter. You just need to know which tiles should be paired. Then you can play these pairs in any order the game permits, and you will win.

If this sounds counterintuitive, consider the following explanation. If you change the order of moves, you are removing certain pairs earlier than in the original solution. Can you get stuck by removing something too soon? No, because removing tiles earlier only makes other tiles available earlier. And can you get stuck by removing something later than in the original solution? No, because all that matters is that you eventually remove it, which allows you to proceed with the following moves. This is the entire logic, and with a bit of skill, it can be turned into a rigorous mathematical proof. We won’t do it here, but from now on, we will treat it as a fact:

All that matters is which tiles are matched. The order in which matched pairs are removed from the board does not matter.

How many ways are there to pair up four tiles? There are three ways to do it (see Figure 3). A solution to the game is thus described by 36 choices, each involving one of three options. If you want to know if it is possible to solve a particular deal, you can check all possible ways to pair tiles. If at least one of them clears the board, then the deal is solvable.

Before we start checking them, let’s calculate the number of such possible pairings. To do it, we use the formula for k-permutations of n with replacements, where n = 3 and k = 36. It turns out that the number of potential solutions to verify is 150,094,635,296,999,121. It is impossible to check them all, even for a single deal, let alone for thousands of them, which would be necessary to estimate the overall fraction of unsolvable deals.

Although the fact we discovered doesn’t help us to determine if a deal is solvable, it leads to two important conclusions. First, whenever you see four available tiles of the same kind, you can remove them immediately. The only way to make a mistake that prevents you from winning is to match the wrong tiles. And if all four of them are available, it does not matter how they are matched, because they are all removed simultaneously. This “sure strategy” is easy to implement, and you should start using it immediately.

The second sure strategy is that after you remove the first pair of a particular kind, the second pair of this kind can be removed as soon as it becomes available. This is because you can make a mistake only during the initial decision which two tiles to remove as the first pair. When you remove the second pair, you are either still on the winning path or already deviated from the solution. This tip is harder to apply than the first one because you need to remember if there are only two tiles left for each kind. However, it can make your game significantly easier and speed it up, especially towards the end, because it gives a green light to the moves that you would otherwise unnecessarily overthink.

Evaluating typical Mahjong strategies

Removing pairs that can’t affect your outcome may seem helpful, but by itself, it will not improve your chances of winning. To increase your win ratio, you need strategies that help you decide which tiles to pair. There are six often-mentioned strategies whose efficacy is easy to test. These are:

1. Maximize the number of available tiles.
2. Maximize the number of pairs that can be removed (i.e., the number of available moves).
3. Focus only on the longest row (other rows don’t matter).
4. Prioritize reducing the lengths of all rows with an emphasis on the longer ones.
5. Prioritize reducing the height of the pile.
6. Think ahead.

Table 1 shows the winning percentages for strategies 1 through 5 (combined with the two sure strategies described in the previous section). To take into account the “think ahead” strategy, a computer played each of the five strategies in two ways: first, by choosing the best move for a given strategy every turn, and secondly, by checking out all sequences of three moves, evaluating which sequence led to the best outcome, and choosing the first move from the best sequence. In essence, for each of the strategies 1 through 5, we are comparing the winning ratio of this strategy without looking ahead to its winning ratio when we look three moves ahead.

Among the five investigated strategies, maximizing the number of available moves is the most effective, followed by focusing on the number of available tiles. Strategies 4 and 5, without thinking ahead, actually perform worse than the baseline of 5.9% achieved by removing pairs randomly. However, this does not mean that these strategies are useless. The best way to play is to combine all five strategies by giving the highest priority to the number of available moves, lower priority to the number of available tiles, and the lowest priority to the row lengths, the longest row, and the pile height. A complex strategy combining our five strategies in this way can achieve a winning percentage of 16.8% without looking ahead, 21.4% if we look three moves ahead, and 24.1% if we look five moves ahead.

This paints a bleak picture: even if we play with inhuman precision and perfect predictive capabilities, we win less than a quarter of the games, let alone the 96% we should be able to win. Of course, we did not include in our calculations all possible “strategies.” Tips, such as focusing on the stacked or adjacent tiles of the same kind, or trying to remove the critical tiles as soon as possible, may further improve our chances of winning. However, the improvements will be modest (a few percentage points) and will not constitute the silver bullet that takes us to the top of our game.

How to win every Mahjong Solitaire game?

The difficulty of the game comes from the fact that Mahjong Solitaire is a game with imperfect information. At the beginning, many tiles are hidden inside the Turtle’s shell. As a result, initially, it is a game of chance – we can’t plan our moves using logic to create a winning strategy, as we would, for example, when solving the Rubik’s Cube. Instead, we try to come up with a mix of strategies that maximizes our chances and hope for a lucky deal.

Nevertheless, there is a way to achieve the 96% winning ratio. To do that, you have to transform Mahjong from a game with imperfect information into a game with perfect information. As soon as you realize that the current deal is hard, stop your typical strategy and start the following ultimate winning strategy.

The first step is to map the pile. This will involve heavy usage of the “Undo” button. You need to take notes – draw a picture of the current deal (on paper or in a spreadsheet – the latter option makes searching easier). Try to disassemble the pile from different directions to get access to every tile. Note their positions. Once you know where all the tiles are, you have the perfect information about the deal and, unless you have an unsolvable deal, you should be able to solve the game with pure logic (and persistence).

As you make failed attempts to solve the puzzle, note which tiles blocked you. Try to determine which should be removed earlier to avoid getting stuck. You can use the picture you made to plan with what to pair the troublesome tiles. The hardest games have multiple “critical” tiles which must be removed in a specific order. Each attempt to solve the game should provide you with additional information and move you closer to victory. This strategy requires determination and patience. If that's what you like, give it a try.

Appendix: methodology

The winning percentages reported in this article were obtained using Monte Carlo simulations, during which a computer program was drawing random deals and attempting to solve these deals using a selected strategy.