Help with this math problem please.

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TheGreatEater

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So if a game as 3 moons that rotate at different speeds, and runs on a 27 day month. And the first moon rotates the planet at once per 27 days. Moon two rotates the earth at once every 81 days, and the last one a game year of 324 days.

How many lunar eclipses will there be, and on what days?

Year: 12 Months [12, 27 day months].

Moon 1: 27 days / rotation: 12 / year.

Moon 2: 81 days / rotation: 4 / year.

Moon 3: 1 rotation / year.
 
TheGreatEater said:
So if a game as 3 moons that rotate at different speeds, and runs on a 27 day month. And the first moon rotates the planet at once per 27 days. Moon two rotates the earth at once every 81 days, and the last one a game year of 324 days.

How many lunar eclipses will there be, and on what days?

Year: 12 Months [12, 27 day months].

Moon 1: 27 days / rotation: 12 / year.

Moon 2: 81 days / rotation: 4 / year.

Moon 3: 1 rotation / year.
Click to expand...

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

As a hint, let's look at a similar problem. Two moons with 11 and 33 day rotations in a 99 day year [9 months of 11 days each]. Assume they are lined up [there is a lunar eclipse] at day 0. At day 33 [the least common multiple of 11 and 33], they will be back in line as they will be on day 66, and 99. So there will be a total of 3 [99 divided by 33] times they are lined up on the 33 day cycle. How about in between those 33 day cycles? Well on day 11, moon 1 has gone through 1 complete revolution but moon 2 is only 1/3 of the way around. On day 22 moon 1 has gone through two complete revolutions but moon 2 is only 2/3 thirds of the way around. Thus moon one lined up with moon two sometime in that 11 days between days 11 and 22. On day 33 the two are aligned again and the cycle starts over. Now since, there was one (33 divided by 11 minus 2) line up (eclipse) totally within that 33 days, that means there will be two (33 divided by 11 minus 1) eclipses in a 33 day cycle [one at the beginning and one in between]. Since there are three 33 day cycles in the year, there will be 2*3=6 lunar eclipses.

For your problem, don't forget the interaction between all of the moon pairs and triplets.
 
Here's what I have so far [and chances are that I'm wrong].

Moon 1: 12 full rotations a year [27 day months. So 1 full rotation. [27 / 27 = 1 [mo]].

Moon 2: 4 Full rotations a year. [81 times 4 = 324 days. Or 1 Rotation very every 3 months. [27 / 81 = 3 [mo]]

Moon 3: 1 Rotation a game year [324 days [27 / 324 = 12 [mo]].
____________

I suck at higher maths, and were I mess up at, is going from months to days.

I know that M1 have two Bi Lunar Eclipses per month [With a 12 month rotation, it will pass by each moon once per month.].

With The first Tri-Lunar being on January. And M2 making 4 rotations at a 3 Month period.

January: First Tri Lunar eclipse [Janurary 1st]. Day 1 - 27 [Moon 3 4/4ths rotation [1 whole]]

February: days 27 - 54 [27*2 = 54]

March: days 54 - 81 [27*3 = 81]

April: Days 81 - 108 [27*4 = 108] [moon 2 full rotation [81*1 = 81]. Moon 3 1/4th rotation around the planet.].

May: days 108 - 135 [27*5 = 135].

June: days 135 - 162 [27*6 = 162]

July: days 162 - 189 [27*7 = 182]] [Moon 2 full rotation [81*2 = 162]. Moon 3 2/4ths around planet.]

August: days 189 - 216 [27*8 = 216]

September: days 216 - 243 [27*9 = 243]

October: days 243 - 270 [27*10 = 270] [Moon 2 full rotation [81*3 = 243] [. Moon 3 3/4ths rotation.]

November: days 270 - 297 [27*11 = 297].

December: days 297 - 324 [12*27 = 324].

Each Eclipse falling a month after normal Seasonal months.

Moon 2: Tri Lunar Months will replace Bi Lunar Eclipses, on above months.

Given a 324 year, though, I'm having problems figuring out which days the moons will intersect. I'll post a diagram of moons, and months they make a full rotation.

moon placement..jpg

But yeah. My total skills with math go up to highschool algebra. Which I haven't really had to use in years. And this is a little math problem that's been bothering me. Since I don't know how to solve it, or where to start. Once I get past the basics of, as I put above.

M1 = 1 rotation every 27 days. [324 / 12]

M2 = 1 every 81 days. [324 / 4]

M3 = 1 every 324 days / 1 year. [324 / 324].

Past that. Janurary 1st is start of first Tri-Lunar Eclipse. I don't know how to figure out what to do to figure out what days M1 will intersect each M / mo. Or when M2 will be part of Tri Lunar Eclipses or Bi Lunar Eclipses. Just what months each one makes a full rotation.
 
Let's approach this a different way. How fast is Moon 1 going? Answer 1 revolution per 27 days. Similary with moon 1 and 2. So
\(\displaystyle s_1\, =\, \frac{1}{27} \frac{rev}{day}\)
\(\displaystyle s_2\, =\, \frac{1}{81} \frac{rev}{day}\)
and
\(\displaystyle s_3\, =\, \frac{1}{324} \frac{rev}{day}\)
In t days, measured from a time when they all line up, the distance traveled from that line up is given by
\(\displaystyle d_1\, =\, s_1\, *\, t\, *\, day\, *\, \frac{rev}{day}\, =\, s_1\, *\, t\, rev\)
\(\displaystyle d_2\, =\, s_2\, *\, t\, *\, day\, *\, \frac{rev}{day}\, =\, s_2\, *\, t\, rev\)
\(\displaystyle d_3\, =\, s_3\, *\, t\, *\, day\, *\, \frac{rev}{day}\, =\, s_3\, *\, t\, rev\)
There will be an eclipse if the distance into the present revolution for each moon is the same.

Lets consider moons 1 and 2. When they both start, moon 1 races ahead of moon two and has gone through a complete revolution while moon 2 has completed only 1/3 of a revolution. Moon 1 keeps on a truckin' and completes two revolutions while moon 2 has completed only 2/3 revolution. Well obviously, moon one caught up with and passed moon 2 sometime during that second revolution for moon one. Let d be the distance into a revolution moon two has traveled. Then, since moon 1 had an additional revolution, it traveled 1+d revolutions. So
1+d = s1 t
d = s2 t
That is two equations in two unknowns so you can solve for d and t for that ellipse. How about the next time around (starting at the end of 81 days or the beginning of the 82 day)? Moon 1 will be back to the beginning of a revolution but so will moon 2 so the process starts all over again. That sequence happens 4 times during the year with 2 eclipses involving moons 1 and 2 for a total of 8 eclipses involving moons 1 and 2.

Now do the same for moons 1 and 3 and for moons 2 and 3. Finally do that for moons 1, 2, and 3 and eliminate duplicates.
 
So (tell me if I'm wrong), but if as you said Moon 1 passes Moon 2 for a total of 8 Eclipses. Then that'd be 324 / 8. Which would give me once every 40 days am I correct? [well 40.5]

If so then on days [40, 81, 121, 162, 202, 243, 283, and 324 are Moon 1 and Moon 2 eclipse dates.].

If each of M2 is 1/4th a rotation of M3. Then ... That mean it passes it 4 times a year right. Then that'd mean each of the M2 full rotations is an eclipse of M2 and M3? And with M1 making 12 rotations then that'd make it happen once every 27 days right? And each of the Full rotations of Moon 3 coincide with those days so. the first of each of those 4 months would be a Tri Lunar Eclipse right?

But yeah I got everything till I saw Sn[which I got] * t * d * days/rev = Sn * t rev. That went over my head. But I'm glad I got that there were supposed to be 8 Eclipses between M1 and M2. With that I was able to judge everything else [since M2 lines about about right with M3 each revolution. And those days on the 40.5 day calculation, match up with dates given as the full rotation of M2's 4 full rotations. [or Days, 81, 162, 243, and 324?]

Did I do it right?
 
-[well 40.5 yes]
-If each of M2 is 1/4th a rotation of M3. Then ... That mean it passes it 4 times a year right. no. 4 times a cycle for M2 IF you count both the beginning and end: M2 & M3 are at the same point in the revolution cycle at the beginning (rotation 1), and during rotation 2, 3, and 4 [where they are both at the start of a revolution again]. It then repeats the number of M2 cycles in a year during the year. If the M2 cycle were one year, then yes it would be once a year.

If so then on days [40, 81, 121, 162, 202, 243, 283, and 324 are Moon 1 and Moon 2 eclipse dates.]. I would have written it as [0, 40, 81, 121, 162, 202, 243, and 283] but yes, you just put the first eclipse during the 40th day and the last one on the 'end of last day' of the year and I put the 1st eclipse on the (beginning of the) first (0th) day and the last one on day 283.

Did I do it right? See comments, but it looks like you got the right idea.
 
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