A277433 Minimal No-3-in-line

59 views
Skip to first unread message

Ed Pegg

unread,
Jun 24, 2026, 10:32:09β€―AMJun 24
to SeqFan
https://oeis.org/A277433Β Β 

14x14 <= 12Β 
{{0, 0}, {0, 13}, {3, 6}, {3, 7}, {6, 3}, {6, 10}, {7, 3}, {7,10}, {10, 6}, {10, 7}, {13, 0}, {13, 13}}
β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡
β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†

19x19<=16Β Β 

β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡

24x24 <= 20

β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡

Ed Pegg

unread,
Jun 26, 2026, 11:04:08β€―PMJun 26
to seq...@googlegroups.com
Someone asked earlier if aΒ comment could be added to a sequence.Β 
https://oeis.org/A005408Β  The odd numbers.Β Β 
Falstaff: "They say there is divinity in odd numbers, either in nativity, chance, or death." Shakespeare, The Merry Wives of Windsor (Act 5, Scene 1).

https://oeis.org/A277433Β Β Martin Gardner's minimal no-3-in-a-line problem, all slopes version.

1, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 10, 12, 12, 14, 15, 16, 16, 16, 18, 20, 20, 20, 20, 22, 24, 24, 24, 24, 28Β  Β 

These are at least upper bounds.Β  I'll pay $20 to anyone that can improve any of the solutions under size 30.Β 



--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/18b339aa-dfd6-4f07-9946-b8b94869b4ban%40googlegroups.com.
MinimalNo3.png

Ed Pegg

unread,
Jun 27, 2026, 1:41:05β€―PMJun 27
to seq...@googlegroups.com
I've put a write-up and interactive code atΒ 
https://community.wolfram.com/groups/-/m/t/3740214Β Β 
along with 101 solutions I've compiled.

Neil Sloane

unread,
Jun 27, 2026, 2:22:34β€―PMJun 27
to seq...@googlegroups.com
Hi Ed!

>Β Β I've put a write-up and interactive code atΒ 
https://community.wolfram.com/groups/-/m/t/3740214Β Β 
along with 101 solutions I've compiled.


Please don't forgetΒ to add a link to your write-up to the entry A277433 (assuming that is the best place for it)!

Best regards
NeilΒ 

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,Β 



To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAGog0%2Bbm97%3D63DvA%2Baf%3DmAyCCnuAN1PtMfQBqo-q7jSaCH71fQ%40mail.gmail.com.

Ed Pegg

unread,
Jun 27, 2026, 3:59:32β€―PMJun 27
to seq...@googlegroups.com
Thanks, Neil.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAAOnSgSqdV_OtCE1LcT_P7c-k_RH36mwJtHBKJG8UG%2BZs46nyw%40mail.gmail.com.

Ed Pegg

unread,
Jun 27, 2026, 6:38:12β€―PMJun 27
to seq...@googlegroups.com
I believe there's a not-too-bad proof of a(12)=10, because it's difficult to get coverage of 144 points with 9 points.Β  If the number of orthogonal lines isn't maximized, coverage seems impossible. Enumerating all the alternating orthogonal chains isn't too bad up to 8-chains. Unfortunately, this method only seems useful for a(12).

orth4[n_] Β := Binomial[n, 2]^2
orth6[n_] Β := 6 Binomial[n, 3]^2
orth8[n_] Β := 72 Binomial[n, 4]^2
orth10[n_] := 1440 Binomial[n, 5]^2
orth[2 k_][n_] := (k! (k - 1)!/2) Binomial[n, k]^2Β  Β 

orth6[12] Β =Β  Β  Β 290,400
orth8[12] Β = Β  Β  17,641,800
orth10[12] = Β  Β 903,260,160

Ed Pegg

unread,
Jun 27, 2026, 8:27:35β€―PMJun 27
to seq...@googlegroups.com
A curiosityΒ ... 15x15 can be covered with 13 points so that adding another point will produce 3-in-a-line ... but it has two 3-in-a-line.Β  Β Β 

β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—†β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—†β—‡β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—†β—‡β—†β—‡β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—†β—‡β—†β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡
β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡β—‡

Ed Pegg

unread,
Jul 2, 2026, 10:25:49β€―AMΒ (12 days ago)Β Jul 2
to seq...@googlegroups.com
Dmitry Kamenetsky supplied improvements for a(16), a(17) and a(21).
MinimalNo3.png

Ed Pegg

unread,
Jul 3, 2026, 2:07:01β€―PMΒ (11 days ago)Β Jul 3
to seq...@googlegroups.com
I just learned ofΒ https://arxiv.org/pdf/2203.13170Β Β 

Geometric Dominating Sets
Oswin Aichholzer, David Eppstein, Eva-Maria HainzlΒ Β 

Β  n 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
InΒ  12 12 14 14 15 16 16 16 16 18 20 20 22 24 24 24 24 25Β Β 

I wasn't too far off.Β  I need to make another update.Β 
n = 20: 18 -> 16
n = 21: 19 -> 16
n = 22: 20 -> 18
n = 30: 28 -> 25

Ed Pegg

unread,
Jul 3, 2026, 6:34:40β€―PMΒ (11 days ago)Β Jul 3
to seq...@googlegroups.com
I thought this would be easy. Easy! I've put in another hack at it.Β Β 
https://oeis.org/A277433Β Β 
https://arxiv.org/abs/2203.13170Β Β 
https://community.wolfram.com/groups/-/m/t/3740214Β Β 
MinimalNo3.png
Reply all
Reply to author
Forward
0 new messages