13 Jun '19 08:57>
@venda
To be unambiguous: Can the “straight line” be planet-planet-sun OR planet-sun-planet?.
To be unambiguous: Can the “straight line” be planet-planet-sun OR planet-sun-planet?.
@iamatiger saidYes, either
@venda
To be unambiguous: Can the “straight line” be planet-planet-sun OR planet-sun-planet?.
@venda saidJust a point on this type of problem. They are deliberately constructed to be confusing. They tend to require fluid intelligence, which is more of a young persons thing, as one ages crystallized intelligence (roughly speaking experience) becomes more important. In so far as they are useful it is because they force one to be careful in analysing data, which in real world problems often isn't that neat and tidy.
At one time I could do this type of thing easily.
Now apparently not because recently I came across this and I can't work out how to get to the answer so I must be going wrong somewhere.The given answer doesn't agree with what I get it to.
I'm sure someone here can solve the puzzle and show me the working out please:-
A man walks from one town to another.
On the first day h ...[text shortened]... urth day half of the remaining distance.He now has 14 miles left.
How far has he travelled?
@deepthought saidI agree.I used to teach my children to look out for traps and also to think about what the answer should be approximately so as to avoid quoting an answer where the working out may be at fault.
Just a point on this type of problem. They are deliberately constructed to be confusing. They tend to require fluid intelligence, which is more of a young persons thing, as one ages crystallized intelligence (roughly speaking experience) becomes more important. In so far as they are useful it is because they force one to be careful in analysing data, which in real world problems often isn't that neat and tidy.
@forkedknight saidWell done.
OK, I think I have a solution.
I found a formula for planetary alignment they way I was assuming (all planets on a single radius away from the sun):
For orbits of period p1, p2, p3, where p1 < p2 < p3:
alignment of p1 with p2:
t = k / (1/p1 - 1/p2), where k is any integer
alignment of p1 with p3:
t = j / (1/p1 - 1/p3), where j is any integer
The three plan ...[text shortened]... half of the radial alignment solution? It doesn't seem quite right to me, but I can't see why not.
@forkedknight saidI was talking about the same alignment occuring after X years,
That's not quite true.
@wolfgang59 saidYes, I got what you meant, but @venda stated in the initial prompt:
I was talking about the same alignment occuring after X years,
where X is the lowest common multiple of the two orbital times.
@iamatiger saidT = FS/(2S-2F)
Two planets that are lined up PPSun will generally line up PSunP in half the time it takes them to get back to PPSun.
If the orbital period of the slow planet is S and the fast planet period is F then the next PSunP will happen when the fast planet has done exactly half an orbit more than the slow planet.
T/F = T/S + 1/2
1/F-1/S = 1/(2T)
So this should be soluble with the same techniques
@iamatiger saidWow this is all deep stuff Tiger.
T = FS/(2S-2F)
What this is saying is that the planets will be in line with the sun whenever
T=JFS/(2S-2F)
Where J is an integer
Using ForkedKnight’s cunning approach we can add a third planet M: M and F will line up when
T = FS/(2S-2F)
What this is saying is that the planets will be in line with the sun whenever
T=KFM/(2M-2F)
and therefore all 3 planets are ...[text shortened]... 1: line up is SMFX
At T9, F=9, M=3, S=3/2: line up is MFXS
At T12, F=12, M=4, S=2: line up is SMFX