Table of Contents

## Card Selection

To begin with, it is useful to understand the card selection algorithm of Threes. Together with other TouchArcade forum members, I reverse-engineered it shortly after the game’s release.

Cards in Threes fall into two categories: *basic cards* (**1**, **2**, **3**) and *bonus cards* (**6**, **12**, **24**, …). The way in which these cards are drawn is quite different. I will therefore cover them separately.

### Basic Cards

The mechanic for drawing basic cards is best described with a face-down stack of cards as one would use in real life card games. In the case of Threes, this stack is comprised of the following cards:

- 4x blue
**1**‘s - 4x red
**2**‘s - 4x white
**3**‘s

As with the physical analogy, card stacks in Threes are also shuffled upon creation. To keep with the analogy, one can imagine the cards having colored backs (to simulate the next card indicator).

When a game starts, a fresh stack is created and shuffled. Then, 9 cards are drawn and placed randomly on the board (when starting with a bonus only 8 cards are drawn in addition to the boost card). With every move, the top card of the stack is drawn which reveals the next card back. When the last card of a stack is drawn after a move, a new stack is instantly created and shuffled. The top card of the fresh stack is then displayed as the next card.

### Bonus Cards

Bonus cards are managed very differently to basic cards. They only come into play when the *high card* (highest card on the board) reaches a certain face value. As the value of the high card increases, higher bonus cards can be drawn. The rules that governs this are as follows:

- The lowest value bonus card is
**6**. - The highest value bonus card is the face value of the high card divided by 8.

In the case that the value of the high card divided by 8 is less than **6**, no bonus cards can be drawn. So, once the high card is **48**, a **6** can be drawn (**48**/8=**6**). From then on, every new rank of high card adds a new possible bonus card to the pool.

Once the pool of bonus cards is no longer empty (high card **48**), for exactly one in 21 moves, the next card will not come from the stack of basic cards but rather from the pool of bonus cards. Unlike the stack of basic cards, the pool of bonus cards is immediately stocked back up after a card has been drawn from it. In other words, each bonus card value is independent of what bonus cards were drawn before. If the next card is a bonus card, up to three consecutive values are displayed as next which have an equal probability of actually being drawn. Each possible triplet of bonus cards is equally likely, meaning that very low and very high value bonus cards become less likely compared to middle-sized bonus cards the larger the pool of available bonus card values becomes.

### Rules of Thumb

Apart from the obvious application of this knowledge, that is to keep track of the cards you draw to infer the current stack and predict future draws, one can draw some rules of thumb:

- If there are four
**1**s (or**2**s) on the board without their counterpart, you will draw them within the next 8 basic cards. - More generally: if the difference between the number of
**1**s and**2**s on the board is 4, the next 4 to 8 draws of basic cards will reduce this difference back to 0. - Outside of these extreme cases, the imbalance between
**1**s and**2**s currently on the board allows you to anticipate what is left on the draw stack.

With this knowledge, you could “count cards” and get a better idea of which cards remain as the stack is depleted. However, the more you play, the more intuition you will develop for this mechanic so that conscious effort is not really helpful.

## Card Placement

While basic and bonus cards differ in the way their face values are generated, the rules that govern their placement on the board are identitcal.

New cards are placed in the row/column opposite of your movement direction. Move left and the new card is placed in the rightmost column, move down and it spawns in the topmost row. Additionally, new cards are only placed in rows/columns that moved, even if there are free spaces in locked rows/columns. If there is more than one eligible spot, each one has an equal chance of being chosen.

By deliberately locking and unlocking rows/columns, you can manage or at least plan around the placement of new cards, which is absolutely vital to achieve higher scores.

## Difficulty Curve

With an ever growing high card, Threes does not only become more difficult/less forgiving due to the decreasing amount of space and increase in the bonus card range but also due to the arrangement of chains towards the high card. Let me illustrate this with examples where the high card is in the lower left corner of the board:

```
6 12
192 96 48 24
```

Notice how the second row is pretty free and bonus cards can easily be shuffled to where they merge well into the chain.

```
6
12 24 48 96
1536 768 384 192
```

Here, the second row is a lot more restricted and bonus cards need to either come at the right time with the right value or be shuffled around a lot in order for them to neatly flow into the chain.

The two examples here are of course used to illustrate the point but the transition is somewhat gradual and potentially easy to miss. At the end of the day, each “stage” (as defined by the current high card) warrants its own approach and board structure.

To summarize: With an increasing high card you not only have less space to work with but also a greater uncertainty regarding the bonus card value and their ideal place in the merge chain. And bonus cards are a necessity, as I will describe below.

## Bonus Cards: A Necessary Evil

As explained above, 12 moves with basic cards gives you exactly 8 **3**‘s (4 natural **3**‘s and 4 pairs of **1**‘s and **2**‘s). The higher your high card goes, the more moves without bonus cards it would take to get to the next high card.

This means that you are almost guaranteed to get (and need) a high bonus card to complete your merge chain. To illustrate this, here’s how many **3**‘s are “contained” in a given card and how many moves without bonus cards it would take to build one from scratch:

Card | # of 3’s | # of moves |

3 | 1 | 1 |

6 | 2 | 3 |

12 | 4 | 7 |

24 | 8 | 13 |

48 | 16 | 25 |

96 | 32 | 49 |

192 | 64 | 97 |

384 | 128 | 193 |

768 | 256 | 385 |

1536 | 512 | 769 |

3072 | 1024 | 1537 |

6144 | 2048 | 3073 |

Now recall that there is exactly one bonus card per 21 moves. High bonus cards are not only an obstacle and a challenge to your overall board structure, but a necessary mechanism to skip ahead in this exponential growth process.

So rather than looking to build an entire merge chain from scratch after increasing your high card, a good board structure is able to use the high value bonus cards that are almost guaranteed to happen on the way to the next high card merge.

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