On May 18, 7:35 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > On May 16, 2:26 pm, timro21 wrote: > > > The Unsolved Problems web site at > > >http://unsolvedproblems.org/ > > > has recently been updated. > > > There are prizes of US$500 for many of the problems listed. > > > Tim > > Hi Tim, > > I think I solved one! > > http://unsolvedproblems.org/index_files/LonelyRunner.htm > > Lonely Runner Conjecture > > Suppose there are k runners, all lined up at the start of a circular > running track of length 1. They all start running at constant, but > different, speeds. > > The Lonely Runner conjecture states that for each runner, there will > come a time when he or she will be a distance of at least 1/k from > every other runner. > > The conjecture has been proved for small values of k (<=7). > > The problem is to prove or disprove the conjecture for the general > case, or for cases where k > 7. > > SOLUTION > -------- > > Express the k runners speeds as rational fractions. > > n1 n2 .. nk > -- -- .. -- > d1 d2 .. dk > > e.g. runner 1 runs n1 laps in d1 seconds > > Where all denominators are odd > and all but one numerator are even. > > Let n1 be the only odd numerator. > > e.g. for k=4 runners > > ODD EVEN EVEN EVEN > --- ---- ---- ---- > ODD ODD ODD ODD > > This still allows arbitrarily high precision > to represent every runner's speed accurately. > > After t = d1 X d2 X ... X dk seconds > All runners are back at the starting line. > > PROOF: at time t each of the runners runs their > time for ni laps (di) multiplied by all other > denominators which is a whole factor. > > The number of laps finished by each runner after > time t can be calculated by the following equations. > > laps1 = n1 X d2 X d3 X ... X dk > laps2 = n2 X d1 X d3 X ... X dk > laps3 = n3 X d1 X d2 X ... X dk > .. > lapsr = nr X d1 X d2 X ... X dk-1 laps > > WHERE > > laps1 is ODD > laps2 is EVEN > laps3 is EVEN > .. > lapsk is EVEN > > PROOF: all factors in the equation for laps1 are odd. > 1 factor in each of the other laps equations is even. > > When the race is at time t/2 > > At time t/2, r1 has run laps1/2 laps. > At time t/2, r2 has run laps2/2 laps. > At time t/2, r3 has run laps3/2 laps. > ... > At time t/2, rk has run lapsk/2 laps. > > laps1/2 has remainder 1/2 > laps2/2 is a whole number > laps3/2 is a whole number > .. > lapsk/2 is a whole number > > This puts runner 1 half way around the track at time t/2 > with every other runner back at the starting position. > > From this it follows that runner 1 is more than > the track length / k distance from all other runners. > > G. Cooper > BINFTECH UQ
Using simple Action-Reaction logic we can prove that there can be no lonely runner in the race.
The umpire of the race who stands in the dead centre of the circular track is the only lonely one in that race.
Alone - Crowded
Singular - Plural
Centre - Perimeter
Because there is only one spot (singular) in the dead centre of a circle, it is the only location where someone can be all alone. There is not enough room for 2 (plural) things.
Official - Competitors
Alone - Crowded
Absolute - Relative
Zero - Value
Lonely is always absolute fixed that equals 0. Implying you have zero people keeping you company.
Relative to the Absolute lonely centre is a degree of crowding present on the perimeter that goes around that centre.
So, if you are to umpire a running race. The first thing you do is walk into the middle of a field where the race will be held and stand in the dead centre. Then you draw a circle around yourself on the ground so that only you can fit inside. You will be all alone during the race, a neutral observer.
Neutral - Engaged
Then you draw another circle around your neutral centre which will be active and crowded with mulitple competitors.
Absolute - Relative
The degree of crowding around the outside is a relative value to the absolute isolation and loneliness of the neutral centre.
Answer - Question
So, next time you have a question about our universe such as:
"Does the moon of our Earth spin on its own axis?"
To solution is easy.
Solution - Problem
Always look to the centre to solve any problem.
You must be a neutral observer in the dead centre of the moon to answer without prejudice whether or not the moon actually spins.
Impartial - Biased
Agree - Disagree
We can only agree with an impartial observer. We always disagree with biased relative opinion.
The moon doesn't spin relative to the fixed stars. The stars (plural) is biased and should be ignored. We can only regard the view of a 'singular' impartial point of reference that is only in the dead centre of the moon. The umpire of the moon.