Solve Word Problem: Average # Lunar Months/Year for Tegummans

[Helios]

Here is a quick word problem to solve!
The Tegummans ( fictional culture ) measure periods of eight years called Octads, for ritualistic purposes. These periods begin and end with a full moon at midnight, and therefore usually contain 99 lunar months, but do seldomly contain 98 months.
The Tegummans also measure periods of 272 years ( 34 Octads ) called Ages. The first Octad of any Age always contains 99 months.
Octads are reduced from 99 months to 98 months every 18th Octad, not counting any first Octad of an Age. That is, the reduction by one month of the 18th Octad is postponed every 34 Octads.
What is the average number of lunar months per year as reckoned by the Tegummans?

 

[mfb]

What does "postponed" mean? Moved to the next Octad?
The Ages and the Octad reduction cycle are both multiples of two. Either they align (then the reduction is skipped sometimes), or they do not (then the exception is irrelevant), or the shifting occurs (then they do not align any more after the first shift happened), or the shifting occurs but does not influence the cycle (then we can have a separate alignment case).

If you fix the Octads to be 8 years long, then you cannot influence the number of lunar months (unless you really screw up the definition of lunar months).

 

[Helios]

Postponed means that in the course of counting the 18th octant ( due for reduction ), should an octant be the first octant of an age ( fixed to 99 months ), that octant is skipped in the counting process resulting in 19 octants from the last correction.
The whole cycle must be computed. Only then can the ratio of months-to-years be evaluated. A year or octant is expressed by the average months it contains.

 

[mfb]

Postponed means that in the course of counting the 18th octant ( due for reduction ), should an octant be the first octant of an age ( fixed to 99 months ), that octant is skipped in the counting process resulting in 19 octants from the last correction.

Then this correction will happen at most once. Afterwards the correction is always in even Octads in the Ages. The Ages become irrelevant and the solution is easy to calculate.

 

[Helios]

Ages are not irrelevant because each age contains a First Octant which is skipped in the counting for the 18th octant. This increases the average between corrections from 18 to a fraction greater than 18.
There is more than one correction in the cycle. Otherwise there would be 1781 months per 144 years, which would be deficient for the requirements of an accurate calendar.

 

[mfb]

Ages are not irrelevant because each age contains a First Octant which is skipped in the counting for the 18th octant.

It will never be reached after it has been reached once. Both the 34 octads and the 18 octads cycles are even. The skipping will always happen in an even octad within an age.

 

[Helios]

It will never be reached after it has been reached once.

What is "It" that will never be reached? The octad which is skipped is the first octad of the age. This changes the effective count to 19 from what would otherwise have been a count of 18, were a first-octad-of-the-age not encountered. In fact, the first octad of the age is odd and the fact that both 34 and 18 are even numbers is not relevant.

 

[mfb]

it = the exception.

Numbers:
Consider an age where we would shorten the first Octad (let's call it octad 1), but do not do this due to the exception rule.
Octad 2 gets shortened.
Octad 20 gets shortened.
Octad 35 starts the second age.
Octad 38 gets shortened.
Octad 56 gets shortened.
Octad 69 starts the second age.
Octad 74 gets shortened.
...

As you can see, the ages always start at odd octads, while the shortened octads are always at even numbers. They never fall together again.

Edit: Ah, you mean the start of an age is not counted for the 18? So instead of 38, we shorten 39? Fine. Then we have a shortened octad every 18*35/34 octads, and the average length of an octad is 99-34/(35*18) months.

 

[collinsmark]

Here is the way I interpret the problem.

Let's start at epoch, Age 1, Octad 1.

The counting (of the 18 Octads) starts at Octad 2, since we're skipping Octad 1, when counting.

So,
The first 98 month Octad occurs at Age 1, Octad 19.
The second 98 month Octad occurs at Age 2, Octad 4.
The third 98 month Octad occurs at Age 2, Octad 22.
The fourth 98 month Octad occurs at Age 3, Octad 7.
... and so on.

So if that's correct, here's the solution:

Spoiler

The calendar has a period of exactly 6 ages. Within that 6 age period there are

• 11 instances of 98 month Octads.
• (6)(34) - 11 = 193 instances of 99 month Octads.
• 204 Octads, total.
• (8)(204) = 1632 years.
  
• A total of (11)(98) + (193)(99) = 20185 months.

Thus 20185/1632 months per year (~ 12.3682598 months/year)

Nifty.

 

[Helios]

Collinsmark has gotten the right answer and understands this calendar!