Azul Tile Counting: The Bag, the Lid, and the Refill

The midgame lesson asked you to count the colors on walls and in pattern lines. This lesson goes one step further, into the bag itself. Azul hides its next tiles in a cloth bag, but the bag holds a known set, it empties at a predictable moment, and it refills from a pile everyone watched being made. That is a lot of information for a game that looks like luck.

The numbers here are worked out from the real component counts: 100 tiles, 20 of each color, 4 tiles per factory display.

What a fresh display holds

Draw 4 tiles from a full bag. For any one color you care about, here is how many of it you will see on that display:

So a given display shows your color about 60% of the time, and usually just one tile. Pairs are common across colors, though: about 80% of displays have at least two tiles of some color, and about 12% have three or four of one color.

Two lessons follow. First, a single tile of a color is the normal offer, which is why the short lines fill so easily and line 5 takes patience. Second, a display with three of a kind is rare, roughly one in eight. When one appears, someone at the table probably wants it. Check who before it reaches them.

Across a whole round, the numbers smooth out. A two-player round deals 20 tiles, so about 4 of each color. The chance that a color does not appear at all on a fresh two-player round is under 1%.

When does the bag run out?

The bag starts with 100 tiles, and each round deals 4 per display. Count it out:

The rulebook’s procedure: when the bag is empty, refill it with the tiles from the box lid and keep filling displays. In the rare case that the lid is empty too, the round starts with the displays as they are.

The last tiles in the bag are knowable

Here is the part most players never notice. The tiles left in the bag before the refill are exactly the tiles that have not been drawn yet. If you know what came out, you know what is left.

At two players the effect is cleanest. Rounds 1 to 4 deal 80 tiles. Round 5 deals the last 20, which are simply the 20 tiles never drawn. If you counted each color at the start of rounds 1 to 4, then round 5’s offer is 20 of that color minus your total. Nothing about it is random.

At four players, round 3 starts with the last 28 tiles from the original bag, then 8 from the refill. At three players, round 4 starts with 16 from the original bag and 12 from the refill.

You do not need to track all five colors. Track the one or two that your long lines need. Four short tallies on a scoresheet is enough.

The refill is short on some colors

When the bag refills, it refills only from the lid. And the lid holds only certain tiles:

• In the lid: leftovers from completed pattern lines, and tiles that fell to floor lines.
• Not in the lid: every tile on every wall, and every tile still waiting in an unfinished pattern line.

So a color that everyone has been placing on their walls is scarce after the refill. A color that has been falling to floors, or clearing out of big lines, is plentiful. The refill is not a fresh bag. It is a bag tilted toward whatever the table did not use.

Every tile that goes to the lid passes in front of everyone, so you can follow its contents roughly as the game goes. A simple read works: which colors have mostly gone onto walls, and which have mostly been thrown away?

Putting the count to work

• Before you start a long line, ask how many of that color are still in circulation. If walls already hold most of them, start a different color.
• Before the refill, plan for a skewed bag. Lines in a color that has been going onto walls may wait a long time.
• At two players in round 5, use your count. If your color has 6 tiles left in the bag and your opponent needs it too, you can see the fight coming before the displays are dealt.
• For denial, a color with few tiles left is easy to squeeze. Taking two of the last five can end someone’s plan for the whole game.

None of this needs a calculator. A few tallies and one question, “where are the tiles of this color right now?”, turn the bag from a surprise into a plan.