OFFSET
1,2
COMMENTS
Upper bounds for a(14) to a(26): 18, 21, 24, 26, 28, 29, 33, 36, 37, 40, 43, 46, 49. These have been obtained from the results of the Al Zimmermann competition. - Dmitry Kamenetsky, Apr 23 2021
Upper bounds for a(15) to a(18): 20, 22, 24, 27. - Fausto A. C. Cariboni, Jul 16 2022
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer New York, 2004, F2, 367-368.
Keith F. Lynch, Posting to Math Fun Mailing List, Apr 02 2016.
LINKS
P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.
Sean A. Irvine, Java program (github)
Dmitry Kamenetsky, Best known solutions for n <= 13.
Dmitry Kamenetsky, 7x7 Golomb Square, Puzzling StackExchange, April 2021.
Matt Parker, Unique Distancing Puzzle.
Samuel B. Reid, The unique solution that causes a(7) to be 7.
Wolfram Demonstration Project, No Repeated Distances.
A. Zimmermann, Al Zimmermann's Programming Contests: Point Packing. (Oct 10, 2009).
EXAMPLE
a(1) is the degenerate case of a single point, a(2)=2 is trivial.
a(3)=3: The points ((1,2),(3,1),(3,2)) have distinct mutual squared distances 1, 4, 5.
a(8)=9 is the first square for which k>n: ((1,1), (1,4), (2,2), (6,1), (7,6), (7,7), (9,2), (9,4)) have 7*8/2=28 mutual squared distances: 1, 2, 4, 5, 8, 9, 10, 13, 17, 18, 20, 25, 26, 29, 34, 37, 40, 41, 45, 49, 50, 53, 61, 64, 65, 68, 72, 73, and no configuration of 8 points fitting on an 8 X 8 square exists.
a(10)=11, only two subsets barring symmetry:
{(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)},
{(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}.
a(11)=13, one of the four subsets of the 12 X 13 grid, barring symmetry: {(0,0), (0,1), (0,9), (0,12), (2,0), (5,3), (6,12), (7,0), (8,4), (10,10), (11,11)}
a(12)=15 is satisfied by {(0,0), (1,0), (1,12), (3,0), (7,0), (7,14), (9,4), (12,11), (13,3), (13,8), (14,2), (14,13)}. - Sean A. Irvine, Jul 13 2020
a(13)=16 is satisfied by {(1,1), (2,2), (2,16), (4,14), (6,14), (7,16), (8,8), (11,2), (11,5), (13,15), (13,16), (16,1), (16,8)}. - Bert Dobbelaere, Sep 20 2020
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Hugo Pfoertner, Aug 06 2011
EXTENSIONS
a(10)-a(11) corrected by Ehit Dinesh Agarwal, May 28 2020
a(12) from Sean A. Irvine, Jul 13 2020
a(13) from Bert Dobbelaere, Sep 20 2020
a(14) from Fausto A. C. Cariboni, Jul 16 2022
STATUS
approved