A few things on this LC.
One. I made a C program that calculated all the permutations one could have. This was the brute force method ... just taking N! / [ m! * (N-m)!] for m from 0 to 88....and summing them up (N is total spaces ... m is number of one color of piece ... N-m is the number of the other color of piece) Worked great when N was small.
I got rounding error as a result, and my answer was wrong, though ... when N got large. I was sad.
The clever way to think about it (and I can't take credit for this logic ... but it makes complete sense), is just taking 2^88 as the number of possible ways to set up the board. (which is what my formula will come up with without rounding errors on the computer)
The big problem will be how neopets defined a year. (In order to calculate the time from these number of possible board set ups)
365. 365.25? 365.2422? 52*7 = 364?
If they really wanted an answer to the nearest year ... you would have to know the number of days in a year to about 20 significant digits. If you found that conversion to 20 signifcant digits ... I'm proud of you.
(for 365 days in a year ... the answer would be
294411158505845765530)
(for 365.25 days in a year ... the answer would be
294209645050331839612)
(Notice how just that small change in days per year makes the 4th digit of the number change)