The generators (S) and (T) of the modular group (SL(2, Z)) transform its fundamental domain (F) into a tessellation of the upper half-plane (H). Every tile in this tessellation is the image of (F) under a word composed of (S) and (T). For example, (STTSTTT(F)) denotes the tile obtained by applying the word (STTSTTT) to (F). If no shorter word produces the same tile, this word is minimal. In this paper, we derive a recurrence relation that counts the distinct tiles associated with each minimal word length. The paper is also available as viXra preprint 2606.0055.

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