- Email: [email protected]
Update on the no-three-in-line problem
JOURNAL.
OF COMBINATORIAL
THEORY,
&k?s
A 27, 365-366 (1979)
Note Update
on the No-Three-in-Line DAVID
Problem
BRENT ANDERSON
University of California at Davis, Davis, California 95616 Communicated by the Managing Editors Received August 23, 1978
Solutions to the no-three-in-line problem found for n = 18, 20, 22, 24, and 26.
containing
2n points have been
For n > 1, let D(n) be the maximum number of points which can be chosen from an n x n square array of points so that no three of the chosen point are collinear. Clearly D(n) < 2n since each row of the square must contain at most two of the points. The no-three-in-line problem is the problem of deciding whether D(n) = 2n for all n > 2. A set of 2n points in the n x n square that has no three collinear will be called a “solution.” It has been shown that D(n) = 2n for 2 < n < 16. In [l] and [7], all solutions for n < 10 are listed. Solutions are given for n = 11 and 12 in [2], for n = 13, 14, 15, and 16 in [4], and for n = 14 and 16 in [8]. A construction due to Erdiis [9] shows that D(p) 3 p for all primes p. This result has been improved in [6], where it is shown that, for all primes p, D(2p) > 3( p - 1). It follows that, given E > 0, D(n) 3 (3/2 - E)TIfor all sufficiently large n. In the other direction, Guy and Kelly [5] have presented a probabilistic argument to support their conjecture that D(n) N cn, where c = (2~r~/3)‘/~ M 1.874. The purpose of this note is to present the results of a computer search for solutions that possess 90” rotational symmetry. Letting R*(n) be the number of solutions with this symmetry, the computer found the results given in Table I. (It is easily seen that R*(n) = 0 for n odd. The values of TABLE
I
n
2
4
6
8
10 12
14
16
18
20
22
24
26
R*(n)
1
1
3
4
7
13
13
7
16
>2
>7
>l
4
365 0097-3165/79/060365~2$02.00/0 All
Copyright &71 1979 by AcademicPress, Inc. rights of reproduction in any form reserved.
366
DAVID
BRENT
ANDERSON
R*(n) for n < 10 can also be obtained from [l].) Sample solutions for n = 18, 20, 22, 24, and 26 are given in the Appendix. APPENDIX:
SOLUTIONS ........ ..o ..O .......... ..o.::a ..... ... .. ..o.....r...o ........ ..o..............o ..... ..0...0 ..........
n.o.0 ......... . ..o.....o............ :a::“:::::::“a:::: . ..o...........o .. ..o........o ...... ... ..o. ......... . ....... a . . 0 ....... .I.....0 0 ................ 00 ......... ..... ..o..o ....... o...........o ::b:.:L::::;:.O ....... 0 ... .. ..o...........o. .. ..rJ.......o .. ..o.....o ............ ....... ..o.o ......
I.
.cJo...............*... . . . . . . . . . . . ..O.....O..
. . . ....... ..o....o .... . ..o ............... :a: .. ..0 .............. .0 ........... ..a..0 b” . . ............ ..o .... . . . ......... ..D.....O. . . . :a::::6.....o ............ ““‘0 ... ..c ).....O ............. ...00. . . . ............ ..o...b ....... . . . ..... . . . o..............o . 9 . ..o.........o ......... ............ . 9 9 ..0...0 a.. .... .o ..O.. ........ III . n-2.2
n=18
..*...*.... ,...cl....*.
:’ *’ %::a:“::::
. ..I. cl..... . . . . ..a.... . . . . . ..a...
). ..,a........
. . . . . . ..“..............U ~:::.b..“..“.O........ ..,....a.......... ..o..**.........*...o... :a:::::“:::“:::*........
.... .... ..0.0................... ‘....b:::....O”...O.... :a:::....a:::::::“:::::: . . . . . . . . . . . ..*o..o...... ::::;:::::::aa::“:::“::: Iv. n=24
:::::a:::a: . ..*....... ..O . . . . . . . 0 . . . . . . . . *so . ..o* . . . . . . ..YV.. s..... . . .. . . ii”: a: ... . . ..o.. ,..... .... .. ... ... ... ...
REFERENCES
1. M. A. ADENA, D. A. HOLTON, P. A. KELLY, Some thoughts on the no-three-in-line problem, in “Combinatorial Mathematics. Proceedings of the Second Australian Conference” (D. Holton, Ed.), Lecture Notes in Mathematics No. 403, Springer-Verlag, New York/Berlin. 2. D. CRAGOS AND R. HUGHES-JONES,On the no-three-in-line problem, J. Combinatorial Theory Ser. A 20 (1976), 363-364. 3. M. GARDNER, Mathematical games, Sci. Amer. Vol. 235 (October 1976), 133-134. 4. M. GARDNER, Mathematical games, Sci. Amer. Vol. 236 (March 1977), 139-140. 5. R. K. GUY AND P. A, KELLY, The no-three-in-line problem, Canad. Math. Bull. 11 (1968), 527-531. 6. R. R. HALL, T. H. JACKSON, A. SUDBERY, AND K. WILD, Some advances in the nothree-in-line problem, J. CombinatoriaI Theory Ser. A 18 (1975), 336-341. 7. PATRICK A. KELLY, “The Use of the Computer in Game Theory,” Master’s thesis, University of Calgary, 1967. 8. T. KLCNE, On the no-three-in-line problem, II, J. Combinatorial Theory Ser. A 24 (1978), i26-127. 9. K. F. ROTH, On a problem of Heilbronn, J. London Mark Sot. 26 (1951), 198-204.
OF COMBINATORIAL
THEORY,
&k?s
A 27, 365-366 (1979)
Note Update
on the No-Three-in-Line DAVID
Problem
BRENT ANDERSON
University of California at Davis, Davis, California 95616 Communicated by the Managing Editors Received August 23, 1978
Solutions to the no-three-in-line problem found for n = 18, 20, 22, 24, and 26.
containing
2n points have been
For n > 1, let D(n) be the maximum number of points which can be chosen from an n x n square array of points so that no three of the chosen point are collinear. Clearly D(n) < 2n since each row of the square must contain at most two of the points. The no-three-in-line problem is the problem of deciding whether D(n) = 2n for all n > 2. A set of 2n points in the n x n square that has no three collinear will be called a “solution.” It has been shown that D(n) = 2n for 2 < n < 16. In [l] and [7], all solutions for n < 10 are listed. Solutions are given for n = 11 and 12 in [2], for n = 13, 14, 15, and 16 in [4], and for n = 14 and 16 in [8]. A construction due to Erdiis [9] shows that D(p) 3 p for all primes p. This result has been improved in [6], where it is shown that, for all primes p, D(2p) > 3( p - 1). It follows that, given E > 0, D(n) 3 (3/2 - E)TIfor all sufficiently large n. In the other direction, Guy and Kelly [5] have presented a probabilistic argument to support their conjecture that D(n) N cn, where c = (2~r~/3)‘/~ M 1.874. The purpose of this note is to present the results of a computer search for solutions that possess 90” rotational symmetry. Letting R*(n) be the number of solutions with this symmetry, the computer found the results given in Table I. (It is easily seen that R*(n) = 0 for n odd. The values of TABLE
I
n
2
4
6
8
10 12
14
16
18
20
22
24
26
R*(n)
1
1
3
4
7
13
13
7
16
>2
>7
>l
4
365 0097-3165/79/060365~2$02.00/0 All
Copyright &71 1979 by AcademicPress, Inc. rights of reproduction in any form reserved.
366
DAVID
BRENT
ANDERSON
R*(n) for n < 10 can also be obtained from [l].) Sample solutions for n = 18, 20, 22, 24, and 26 are given in the Appendix. APPENDIX:
SOLUTIONS ........ ..o ..O .......... ..o.::a ..... ... .. ..o.....r...o ........ ..o..............o ..... ..0...0 ..........
n.o.0 ......... . ..o.....o............ :a::“:::::::“a:::: . ..o...........o .. ..o........o ...... ... ..o. ......... . ....... a . . 0 ....... .I.....0 0 ................ 00 ......... ..... ..o..o ....... o...........o ::b:.:L::::;:.O ....... 0 ... .. ..o...........o. .. ..rJ.......o .. ..o.....o ............ ....... ..o.o ......
I.
.cJo...............*... . . . . . . . . . . . ..O.....O..
. . . ....... ..o....o .... . ..o ............... :a: .. ..0 .............. .0 ........... ..a..0 b” . . ............ ..o .... . . . ......... ..D.....O. . . . :a::::6.....o ............ ““‘0 ... ..c ).....O ............. ...00. . . . ............ ..o...b ....... . . . ..... . . . o..............o . 9 . ..o.........o ......... ............ . 9 9 ..0...0 a.. .... .o ..O.. ........ III . n-2.2
n=18
..*...*.... ,...cl....*.
:’ *’ %::a:“::::
. ..I. cl..... . . . . ..a.... . . . . . ..a...
). ..,a........
. . . . . . ..“..............U ~:::.b..“..“.O........ ..,....a.......... ..o..**.........*...o... :a:::::“:::“:::*........
.... .... ..0.0................... ‘....b:::....O”...O.... :a:::....a:::::::“:::::: . . . . . . . . . . . ..*o..o...... ::::;:::::::aa::“:::“::: Iv. n=24
:::::a:::a: . ..*....... ..O . . . . . . . 0 . . . . . . . . *so . ..o* . . . . . . ..YV.. s..... . . .. . . ii”: a: ... . . ..o.. ,..... .... .. ... ... ... ...
REFERENCES
1. M. A. ADENA, D. A. HOLTON, P. A. KELLY, Some thoughts on the no-three-in-line problem, in “Combinatorial Mathematics. Proceedings of the Second Australian Conference” (D. Holton, Ed.), Lecture Notes in Mathematics No. 403, Springer-Verlag, New York/Berlin. 2. D. CRAGOS AND R. HUGHES-JONES,On the no-three-in-line problem, J. Combinatorial Theory Ser. A 20 (1976), 363-364. 3. M. GARDNER, Mathematical games, Sci. Amer. Vol. 235 (October 1976), 133-134. 4. M. GARDNER, Mathematical games, Sci. Amer. Vol. 236 (March 1977), 139-140. 5. R. K. GUY AND P. A, KELLY, The no-three-in-line problem, Canad. Math. Bull. 11 (1968), 527-531. 6. R. R. HALL, T. H. JACKSON, A. SUDBERY, AND K. WILD, Some advances in the nothree-in-line problem, J. CombinatoriaI Theory Ser. A 18 (1975), 336-341. 7. PATRICK A. KELLY, “The Use of the Computer in Game Theory,” Master’s thesis, University of Calgary, 1967. 8. T. KLCNE, On the no-three-in-line problem, II, J. Combinatorial Theory Ser. A 24 (1978), i26-127. 9. K. F. ROTH, On a problem of Heilbronn, J. London Mark Sot. 26 (1951), 198-204.