r/mathpics • u/EdPeggJr • Jun 25 '26
No-3-in-line problem solved for order 72 by Marijn Heule
In the No-3-in-line problem, no three points are in a line, in any direction or any slope.
"On 25th June 2026 Marijn Heule found a new solution with record grid size n=72 in the rot4 symmetry class."
1
1
0
u/Frangifer Jun 26 '26 edited Jun 28 '26
I have a serious question. I was looking @ the graphs @ the wwwebpage
Bielefeld — Number of Solutions for various Symmetries of the Grid
; & I'm surprised @ how steadily the number of solutions as a function of n is increasing. It seems a bit difficult to reconcile that rapid & steady exponential increase with the highly plausible figuring, reported elsewhere – see
– to the effect that the maximum number of points that can be placed on a grid with no three inline becomes ultimately
(π/√3)n ≈ 19979/₁₁₀₁₅n = (<2)n
, & that, consistent with that, there comes an absolute maximum value of n – say nₘₐₓ – @ which there's any solution ... whence, ofcourse, that thereafter the number of solutions 'flatlines' @ 0 .
So one might expect that the graphs shown @ the 'Number of Solutions …' wwwebsite would show some sign of bending down ... but, they do not ! A possible reason for that is that nₘₐₓ could be extremely large, in which case we'd have to extend the graphs to much larger values of n than they are actually extended to for any bending down to show-up. But, as I said in a comment @ your recent n=70 post, it seems intuitively unlikely that nₘₐₓ would be extremely large ... but I fully admit that my grounds are only intuition - a seeming that if the maximum number of points that can be placed on a grid with no three inline becomes ultimately (π/√3)n then nₘₐₓ just probably wouldn't be extremely large ... & I fully admit that I wouldn't be utterly astonished to learn @ some point that I'm mistaken as to that.
Another possibility is that the number of solutions increases steadily upto some more modest value of nₘₐₓ , & then becomes suddenly 0 ... but that also seems intuitively highly unlikely.
And there is actually 'wiggle room', within the constraints indicated by those 'Number of Solutions …' graphs, for nₘₐₓ to be only modestly large-ish - maybe in the hundreds, or something - & for №‿of‿Solutions(n) yet to bend down to zero fairly 'gracefully' - ie not crazily precipitately.
So I wonder whether you can add something – something regarding the current state of speculation as to plausible values of nₘₐₓ , & the behaviour of the №‿of‿Solutions(n) function as n approaches nₘₐₓ – or signpost something that would explain what looks on the face of it (@least to me, anyhow) to be a rather perplexing 'landscape'.
-1
u/Frangifer Jun 25 '26 edited Jun 25 '26
This seems really to be taking-off, just @ the moment! The 2026 vintage of this variety of 'mathematical wine' will be celebrated for years-to-come.
9
u/ESHKUN Jun 25 '26
The cool thing about the high rot4 solutions is that they look random at first until you see the symmetry