[Solved]: Infer number of comets, based upon observations of their periodic visits to Earth

Problem Detail: 

Can we algorithmically infer the number of comets that orbit the Earth, based upon periodic observations of them, if we cannot tell the comets apart? In more detail:

The problem

• There is an unknown, fixed number of comets, numComets that pass by Earth and are visible from the ground.

• Each comet takes a fixed number of years to orbit Earth.

• Based only on this knowledge and the dates recorded that a comet was sighted in the night sky, determine numComets, AND determine the orbital period of each comet. You can use an unlimited amount of observation data (time), but the solution in the shortest amount of
  time is preferable.

Assumptions you can make

• All comets look identical from the ground; there is no way to visually identify them.

• No other object will be mistaken for a comet, and all comets will be seen.

• No comet takes more than 100 years to orbit Earth, and no new comets will be introduced.

As current answers have pointed out, the problem as it lay now is unsolvable. Would a solution be possible if numComets is known? (Even roughly get the right answer?) What modifications to the problem would have to be made otherwise, to still encasulate the spirit of the problem, and make it solvable? (or is it dead with no hope of solving accurately?)

Further assumption

All comets start at a different point in their orbit. I.e. They do not all start their orbit at the same place like racehorses coming out of a gate.

Clarification

Comets are only recorded once per year. So there is no difference between a comet that passes at the beginning of a year and at the end. The problem could just as easily have been worded as days.

Asked By : Airhead

Answered By : lPlant

Based on the restriction you gave, that you cannot identify one comet from another, there is no difinative method of calculation. Here is a proposed situation, There are 2 comets following the same trajectory, they each orbit at exactly ten year orbit durations, so each one is seen every ten years, they are however exactly 5 years apart in their orbits. From the ground if we cannot differentiate between these two comets we only see a comet appear overhead regularly every 5 years, there is no way to tell if this is one comet appearing ever 5 years, or two comets appearing every ten years, but 5 years apart. From this it is impossible to deterministically say how many comets there are, so no algorithm can give you numComets with the assumptions that you have specified.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/28832