If you figure this out, you have solved P v NP
This would no longer be an old joke:
Humor: <PERSON> and <PERSON> Play Marbles
<PERSON> begins, “You know, <PERSON>, people say we think so much alike we must be twins, but we are obviously different ages.” <PERSON> responds, “Yeah, I know, it sort of a twin paradox – I think <PERSON> started that one. Forget it, let’s play marbles.” <PERSON> said, “Here’s the rules: From a 50/50% mix of red and blue marbles we each have to put the blue marbles in a blue container and the red marbles in a red container.” <PERSON> responded, “I’ve done something like this before, and I noticed I get worse and worse as the number of items to sort gets larger and larger.”
Being the mathematicians they are, they decided to give each other a 50/50% chance of winning. That meant doing some preliminary games to develop a handicap – like in golf. First <PERSON> tried it. Given 2 marbles to sort, he completed the task in 2 seconds. 4 marbles, 4 seconds; 10 marbles, 10 seconds, etc. 406
Now, for <PERSON>: 2 marbles, 2 seconds; 4 marbles, 24 seconds; 10 marbles 3628800 seconds, etc. <PERSON> said, “Looks like I scale linearly as x, as the number of marbles, increase … But, wow, <PERSON>, you scale as a factorial, x!”. They both realized the handicap was x! / x, with <PERSON>’s nonpolynomial rate (x!) on top of <PERSON>’s simple polynomial rate (x).
<PERSON> said, “I think I know how to normalize for our respective handicap using a new vehicle I just invented that runs on an E=mc^2 engine and dilates time (t) to t’ based on division of t by that equation you just discovered with ( 1 – (v^2 / c^2))^-0.5. We will have to neglect F = ma because it will be crushing and W = fd because of the consequential heat at the launch site.”
<PERSON> agreed and added, “Let’s play for the best rate, after normalization, because I don’t want to do all the integrals for total velocity and time!” <PERSON> fret that he had problems naming the vehicle, “I found that Folks-Wagon is taken, so I called it a You-Van. In fact, I’ll ride first because you are so slow.” They worked out the math and found the necessary velocity for the handicap to sort 100 marbles:
t’ = t (( 1 – (v^2 / c^2 ))^-0.5) solve for v, with c = 1: v = ( t’^2 – t^2)^0.5) / t’ substitute the scaling factors: v = (((x!)^2 – x^2)^0.5) / (x!) enter actual numbers: v = (((100!)^2 – 100^2)^0.5) / (100!)
<PERSON> looked again and said, “Too bad <PERSON> and <PERSON> haven’t been born yet!” <PERSON> added, “Oops, we have to wait for <PERSON>, too, as we need a lot of precise digits for this calculation. Another day …”
In other words: As Velocity Approaches Light Speed, P Becomes Equivalent to NP for Computations Using Zero-Mass Particles 489