A193839 -id:A193839 - cp's OEIS frontend

A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16, 18

Offset: 1

Hugo Pfoertner, Aug 06 2011

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

			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

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.

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).

Cf. A193839, A003022.

See A271490 for the inverse function.

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

A303331 a(n) is the minimum size of a square integer grid allowing all triples of n points to form triangles of different areas.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 9, 11, 15, 19

Offset: 1

Bert Dobbelaere, Apr 30 2018

Place n points on integer coordinates of a square grid of dimension L x L, so that no 3 points are collinear and the areas of the triangles formed by the binomial(n,3) possible triples are all different. a(n) is the minimum L for which this can be achieved. (Note that there are (L+1)^2 lattice points.)
The fact that all triangle areas are multiples of 1/2 and the maximum area supported by the grid is L^2/2 provides us with a lower bound for a(n).
The problem can be considered a 2-dimensional extension of a Golomb ruler and scales to higher dimensions. In general, consider k points on integer coordinates of a D-dimensional hypercube, forming binomial(k,D+1) D-simplices of which the volumes are all different. For D = 1, we recognize the Golomb ruler; for D = 2, we have the problem defined above.
Just like for Costas arrays (another 2-D extension of Golomb rulers), no 2 displacement vectors can be identical, as the diagonal of a parallelogram cuts the shape in triangles of identical area.
a(11) <= 24, a(12) <= 29. - Hugo Pfoertner, Nov 05 2018

			For n = 5, a solution satisfying unequal triangle areas is {(0,4),(1,1),(3,0),(3,3),(4,3)}, which can be verified by considering the binomial(5,3) = 10 possible triangles by selecting vertices from this set. Each coordinate is contained in the range [0..4]. No smaller solution is possible without creating areas that are no longer unique, hence a(5) = 4.
From _Jon E. Schoenfield_, Apr 30 2018: (Start)
Illustration of the above solution:
                        vertices   area
  4   1  .  .  .  .     --------   ----
                          1 2 3     2.5
  3   .  .  .  4  5       1 2 4     4.0
                          1 2 5     5.5
  2   .  .  .  .  .       1 3 4     4.5
                          1 3 5     6.5
  1   .  2  .  .  .       1 4 5     0.5
                          2 3 4     3.0
  0   .  .  .  3  .       2 3 5     3.5
  y                       2 4 5     1.0
    x 0  1  2  3  4       3 4 5     1.5
(End)

IBM Research, Ponder This Challenge October 2018. Similar problem for rectangular grids.

For optimal Golomb rulers, see A003022.

Cf. A193838, A193839, A271490, A322740.

def addNewAreas(plist, used_areas):
    # Attempt to add last point in plist. 'used_areas' contains all (areas*2)
    # between preplaced points and is updated
    m=len(plist)
    for i in range(m-2):
        for j in range(i+1,m-1):
            k=m-1
            p,q,r=plist[i],plist[j],plist[k]
            # Area = s/2 - using s enables us to use integers
            s=(p[0]*q[1]-p[1]*q[0]) + (q[0]*r[1]-q[1]*r[0]) + (r[0]*p[1]-r[1]*p[0])
            if s<0:
                s=-s
            if s in used_areas:
                return False # Area not unique
            else:
                used_areas[s]=True
    return True
def solveRecursively(n, L, plist, used_areas):
    m=len(plist)
    if m==n:
        #print plist (uncomment to show solution)
        return True
    newlist=plist+[None]
    if m>0:
        startidx=(L+1)*plist[m-1][0] + plist[m-1][1] + 1
    else:
        startidx=0
    for idx in range(startidx, (L+1)**2):
            newlist[m]=( idx/(L+1) , idx%(L+1) )
            newareas=dict(used_areas)
            if addNewAreas(newlist, newareas):
                if solveRecursively(n, L, newlist, newareas):
                    return True
    return False
def A303331(n):
    L=0
    while not solveRecursively(n, L, [], {0:True}):
        L+=1
    return L
def A303331_list(N):
    return [A303331(n) for n in range(1,N+1)]
# Bert Dobbelaere, May 01 2018

a(n+1) >= a(n) (trivial).

a(n) >= sqrt(n*(n-1)*(n-2)/6) for n >= 2 (proven lower bound).

A351699 T(n,k) is the number of non-congruent maximal subsets of a grid of n X k lattice points (k <= n), such that no two points are at the same distance, and the points do not fit into a smaller grid. The size of the subsets is given by A351700. T(n,k) and A351700 are triangles read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 5, 10, 1, 5, 28, 7, 21, 2, 19, 8, 104, 330, 2, 1, 4, 70, 15, 110, 574, 1, 3, 30, 272, 205, 4, 71, 563, 1991, 4, 68, 50, 1001, 113, 1130, 4, 76, 383, 9, 8, 362, 35, 1150, 23, 363, 3975, 7, 38, 8, 18, 1082, 415, 2, 638, 7503, 23, 515, 5802, 2, 2, 150, 62, 4238, 120, 1, 55, 1776, 17277, 26, 481, 2388

Offset: 1

Rainer Rosenthal and Hugo Pfoertner, Apr 09 2022

Configurations of points differing by any combination of rotation and reflection are counted only once.

			The triangle begins:
  #
  # 1:  1                   Counting grids n X k.
      ( 1 )                 Two lines per side length n:
  # 2:  2  2                1. for other side k = 1, 2, ...
      ( 1  1 )                 maximal number of points
  # 3:  2  3    3           2. number of configurations
      ( 1  2    1 )
  # 4:  3  4    4    4      Example: 28 figures with
      ( 1  1    5   10 )             4 points on 5 X 3
  # 5:  3  4    4    5    5
      ( 1  5   28    7   21 )
  # 6:  3  4    5    5    5    6
      ( 2 19    8  104  330    2 )
  # 7:  4  5    5    6    6    6    7
      ( 1  4   70   15  110  574    1 )
  # 8:  4  5    5    6    7    7    7     7
      ( 3 30  272  205    4   71  563  1991 )
  # 9:  4  5    6    6    7    7    8    8   8
      ( 4 68   50 1001  113 1130    4   76 383 )
  #10:  4  6    6    7    7    8    8    8   9    9
      ( 9  8  362   35 1150   23  363 3975   7   38 )
  #11:  4  6    6    7    8    8    8    9   9    9 10
      ( 8 18 1082  415    2  638 7503   23 515 5802  2 )
  #
  #   Grid n X k configurations with
  #       distinct distances
  .
  .
  All T(6,3) = 8 configurations
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  X  X  .  X  .               2 |  .  .  .  .  X  .
      1 |  .  .  .  .  .  X               1 |  .  .  .  .  .  X
      0 |  X  .  .  .  .  .               0 |  X  .  X  .  .  X
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,9,10,17,20,26}  dist^2   {1,2,4,5,8,9,10,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  X  .  X  .               2 |  .  X  .  X  .  .
      1 |  .  .  .  .  .  X               1 |  X  .  .  .  .  .
      0 |  X  X  .  .  .  .               0 |  X  .  .  .  .  X
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,10,13,17,20,26}  dist^2  {1,2,4,5,8,10,13,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  .  .  X  .               2 |  .  .  X  .  X  .
      1 |  X  .  .  .  .  X               1 |  X  .  .  .  .  X
      0 |  X  .  X  .  .  .               0 |  X  .  .  .  .  .
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,10,17,20,25,26}  dist^2  {1,2,4,5,8,10,17,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  X  .  .  X               2 |  X  .  .  .  .  X
      1 |  .  .  .  .  .  .               1 |  .  .  .  .  .  .
      0 |  X  X  .  .  .  X               0 |  X  .  .  X  X  .
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,4,5,8,9,13,16,20,25,29}  dist^2  {1,4,5,8,9,13,16,20,25,29}
  .

Hugo Pfoertner, Table of n, a(n) for n = 1..91, rows 1..13 of triangle, flattened

Cf. A193838, A193839, A271490, A325677, A335232, A351700, A351701.

Completed row 8 and new rows 9-12 from Hugo Pfoertner, Jul 12 2022

A335232 Number of largest subsets of the set of points in an n X n square grid, such that no two points are at the same distance.

Original entry on oeis.org

1, 6, 40, 184, 280, 16, 8, 26800, 4376, 416, 16, 27488, 536, 587640, 10192, 128

Offset: 1

Ehit Dinesh Agarwal, May 27 2020

Ehit Dinesh Agarwal, C program
P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.
Matt Parker, MPMP: Unique Distancing Problem, Youtube video, May 2020.
Giovanni Resta, The essentially unique 7 x 7 configuration

Cf. A193838, A193839.
Size of largest subset in A271490.
Cf. A351699 (generalization to grid rectangles, counting congruent configurations only once).

a(14)-a(16) from Fausto A. C. Cariboni, Jul 03 2022

A193555 Numerators of the squared radii of the smallest enclosing circles of n points with integer coordinates and distinct mutual distances, arranged such that the radius of their enclosing circle is minimized. Denominators are given in A193556.

Original entry on oeis.org

1, 5, 5, 5, 5365, 205, 1885, 117925, 3445, 97, 2225, 62530, 284345, 461, 146605

Offset: 2

Hugo Pfoertner, Jul 30 2011

Finding optimal solutions of this problem has been the topic of a round of Al Zimmermann's programming contests from July to October 2009, entitled "Point Packing".

Conjectured next terms are a(17)/A193556(17)=19720/121, a(18)/A193556(18)=5002/25.

P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.
H. Lefmann and T. Thiele, Point sets with distinct distances, Serie B Informatik, B 94-16, 1994.
H. Lefmann and T. Thiele, Point sets with distinct distances, Combinatorica (1995) 15: 379.

Cf. A193556 (corresponding denominators), A193839.

Cf. A193838 (similar problem for smallest enclosing square).

cp's OEIS Frontend

A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen.

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A303331 a(n) is the minimum size of a square integer grid allowing all triples of n points to form triangles of different areas.

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A351699 T(n,k) is the number of non-congruent maximal subsets of a grid of n X k lattice points (k <= n), such that no two points are at the same distance, and the points do not fit into a smaller grid. The size of the subsets is given by A351700. T(n,k) and A351700 are triangles read by rows.

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A335232 Number of largest subsets of the set of points in an n X n square grid, such that no two points are at the same distance.

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Author

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Extensions

A193555 Numerators of the squared radii of the smallest enclosing circles of n points with integer coordinates and distinct mutual distances, arranged such that the radius of their enclosing circle is minimized. Denominators are given in A193556.

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