# Guy's Twenty Questions in Sylver Coinage

In the American Mathematical Monthly for October 1976, Richard K. Guy propounded twenty unanswered questions about Sylver Coinage. Here is their status as of 2001-02-07:
1. Is there an effective technique for computing the status of a general position M? % Still unknown. For all anybody knows, the question may be undecidable.
2. Is there an effective technique for producing good replies when such exist? % Still unknown. Currently this requires infinite labor.
3. What is the status of position {n} for n of the form 2a ⋅ 3b? % Still unknown for n>12. In particular:
4. What is the status of {16}? % Unknown. Sufficiently large replies divisible by 2 and not by 4 produce short positions, but these positions have even moves that produce new short positions. It is not likely that a winning move can be found.
5. What is the status of {18}? % Unknown. As with {16}, second moves that produce short positions allow many third moves that produce new short positions. I had hoped that {18,22} would be P, but it loses to 79.

It is conjectured that 24 is a good reply to 16 and that 12 and 27 are good replies to 18. [J. H.] Conway offers \$50 for the first establishment of the status of a number 2a ⋅ 3b ≥ 16. % The first two conjectures are false; {10,16,24} and {10,12,18} are P. The status of {18,27} is unknown.

6. Is M a P-position whenever 2M is a P-position? % It usually is not. For example, let M={5,6,9}.
7. Is M a P-position whenever 3M is a P-position? % No. No generalizations can be made about positions with g=3, but for a counterexample let M={4,5,6}.
8. If the game is played “between intelligent players” is it always the case that the first person to make the game bounded is the loser? % Guy goes on to say that the question is not completely well posed. What is intelligent play by the losing player? In any case we need to know more about question 3.
9. Is there a winning strategy of bounded length? % Same difficulties.
10. Is there an N-position with g>1 for which all good replies lead to positions with g=1? % Yes, most of those where g=2. For example, the only winning moves in {8,18,28,38} are 27 and 39.
11. Is G({4}) = ω + 1? % Further calculation has eliminated reasonably low finite values. Nobody has proved that they can all be eliminated.
12. What is G({6})? % This is even harder than 11.
13. What is the least ordinal which is not the G-value of any Sylver Coinage position? The answer is at most ω2. % Unknown.
14. In {4,a,b}, does b/a tend to a limit? % Unknown.
15. In {5,a,b}, does b/a approach 1 for {5,a,2a,b,2b}? % Unknown.
16. In {5,a,b}, does b/a approach 1 for {5,a,2a,b,a+b}? % Unknown.
17. In {5,a,b}, does b/a approach 3 for {5,a,2a,3a,b}? % Unknown.
18. In {5,a,b}, is |ab|=1 or 2 for {5,a,2a,b,2b}? % I do not remember whether a counterexample has been found.
19. Are there infinitely many {5,a,b} for {5,a,2a,3a,b}? % Unknown.
20. Compute the winning moves in all positions made up of numbers up to 12. % The unsolved cases were: {7,12} ; {8,9} ; {8,11} ; {8} [12,14]; {9,10} ; {9,11} [13,30]; {10,11,12} [9,14,15,17]; {10,11} [24,28,47]; {10,12} [7,18]; {10} [5,14,26]; {11,12} ; {12} [8,...]; { } [p,...], where p is a prime greater than 3, and at most finitely many composites 2m ⋅ 3n, where m and n are nonnegative. See also the List of {m,n} Positions.

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Col. G. L. Sicherman [ HOME | MAIL ]