This question comes from ancient India. In Sanskrit poetry, there are short syllables (S) and long syllables (L). The short syllables last one unit of duration, while the long syllables last two units. The patterns of syllables SSS, SL, and LS all last three units. How many patterns of syllables last $11$ units? Show hint

As usual it helps to try a simpler problem first, maybe $4$ or $5$ units. How many patterns are there that start with S? How many with L?

For $4$ units, there are $5$ patterns:

SSSS |

SSL |

SLS |

LSS |

LL |

For $5$ units there are $8$ patterns:

SSSSS |

SSSL |

SSLS |

SLSS |

SLL |

LSSS |

LSL |

LLS |

There are $5$ starting with S and $3$ starting with L. If a $5$-unit pattern starts with S, there are $4$ units left to fill. We already know that there are $5$ patterns that last $4$ units, so it makes sense that there are $5$ patterns starting with S. The patterns that start with L have $3$ units left to fill, so there are $3$ of them (the number of $3$-unit patterns).

Now we have a quick way to work it out. For patterns lasting $6$ units, there are $8$ starting with S and $5$ starting with L, for a total of $13$. For $7$ units, $13 + 8 = 21$ patterns. This is the Fibonacci sequence (first explored in Question 3.1.4), where every term is found by adding the previous two.

Units | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ |

Patterns | $1$ | $2$ | $3$ | $5$ | $8$ | $13$ | $21$ | $34$ | $55$ | $89$ | $144$ |

Hence there are $144$ patterns of syllables with a duration of $11$ units.