Quad>
"A framework of Rogers-Ramanujan identities and their arithmetic properties"
http://arxiv.org/pdf/1401.7718v2.pdf
Beware, there's a good amount of patent-office algebra.
Cheers,
Robert
----------
Not so bad--you can tell it's about numbers. I'm sure the math is
excellent, but I'm "surprized" (p2) that Ken Ono would put his name
on a paper with G(e^-(2π)) written in terms of
(1 + 3*Sqrt[5] - 2*Sqrt[10 + 2*Sqrt[5]])^(1/4)
(the 4th root of a trinomial, also p2) when Corey's denester gives
Sqrt[Sqrt[2] 5^(1/4) - Sqrt[1 + Sqrt[5]]]
(the square root of a binomial), proving my remark to Gary Antonick
just the other day that automatic radical denesting is still in a
sorry state.
--rwg
Here's another game-related problem. Consider the game of "othello"
(aka "reversi"
which has the same rules but the other start position; 8x8 board,
bicolored disks,
start with 2 disks of each color in the central 4)
Q1. what is the maximum "mobility" i.e. number of legal moves, in an
othello position?
Q2. same question, but position must be reachable from gamestart
position via legal move sequence. (Hence the answers for Q2 might
differ for reversi vs othello.)
Q3. What is the shortest possible "wipeout" game (ends when one side
has no disks)?
(Again, answers for Q3 might differ for othello vs reversi.)
Q4. What is the shortest possible nonwipeout game (ends when both
sides have no moves)? (Again, answers for Q4 might differ for othello
vs reversi.)
Q5. Sid Cox conjectured any fully-filled-board (final) position
reachable from the reversi start position, also is reachable from the
othello start position.
Q6. Aubrey de Grey showed that the maximum possible number of times a
disk could be flipped is given by the following table, in an othello
game:
0 1 2 3 3 2 1 0
1 17 18 17 17 18 17 1
2 18 20 20 20 20 18 2
3 17 20 23 23 20 17 3
3 17 20 23 23 20 17 3
2 18 20 20 20 20 18 2
1 17 18 17 17 18 17 1
0 1 2 3 3 2 1 0
and all these upper bounds are tight EXCEPT perhaps the four 17s
located adjacent to the corners, for which he proved a lower bound of
16. Open question: 16 or 17?
A1. As opening bids, 24 is achievable using rows BWooWBWo.
Better: 33 is achievable:
oooooooo
WWWWWWWo
BWoBBBWo
oooooBWo
oWWWoBWo
oWBWooWo
oWWWoWWo
oooooBWo
I suspect 40 is a (probably weak) upper bound.
Neither of these positions is reachable.
A3.
In Martin Gardner's Scientific american column (April 1960?), for Reversi,
1. d5 2. e5 3. d4 4. e4 (these 4 are the opening position, now
for genuine moves:)
5. f4 6. e3 7. f2 8. c4 9. b4 10. c5 11. d6
is an 11-0 wipeout. This final position is not reachable from the
othello start.
For othello,
1. d5 2. e5 3. e4 4. d4 (these 4 are the opening position, now for
genuine moves:)
5. e6 6. f4 7. e3 8. f6 9. g5 10. d6 11. e7 12. f5 13. c5
and
5. d3 6. c5 7. e6 8. d2 9. c4 10. f5 11. c6 12. b5 13. d6 14. d7
(13-0 and 14-0 wipeouts) were found by Manabu Maruo and David Haigh.
I think these are claimed to be proven shortest possible.
A4. Aubrey de Grey found an othello game ending with 15-4 score which
he conjectures is shortest.
http://www.britishothello.org.uk/killball.pdf
--
Warren D. Smith
http://RangeVoting.org
