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### Solutions for Act IV Scene 2 - Tiles

The tiles in the image are from the game mahjong, specifically those from the dots suit. A quick observation is that the number of dots on each tile is in ascending order, hinting that we need to fill the blank tiles in the same way. In order to determine how to fill in the tiles, however, requires the use of a logic leap and a little research:

• The rightmost numbers indicate the total number of dots in each row
• In mahjong, there are only four of each tile

By satisfying these two rules, the blank tiles can be filled in a unique way (possibly via some min/maxing) to give:

 1112366667899 1114566789999 1122223467888 1122233334445 1223333555567 1234444567777 2223334445999 2225788889999 6677778889999

Now that we've filled in the tiles, what do we do with them? There are 13 tiles per row, but in mahjong (or, at least, in most common variants of mahjong) 14 tiles are needed in order to win. Hence the next step is to find the tiles which will allow us to form a valid completed hand, consisting of 4 sets of three (either a triplet like 555 or a sequence like 123) and a pair. For example, 1223333555567 can win off 1, 2 and 4, as it can be broken down into 123 23 33 555 567 (winning off 1/4), or 123 2 333 555 567 (winning off 2).

HandWaits
1112366667899149
111456678999936
1122223467888158
112223333444514
1223333555567124
12344445677771
222333444599923456
22257888899995
6677778889999568

Summing the dots of each group of wait tiles (akin to the start of the puzzle) and converting to letters gives the answer NINE GATES, which is the mahjong hand 1112345678999 capable of winning from all of 123456789.

The answer is: ninegates