Google recently showed how AI could find new solutions for old problems.
But what is the exact answer? The google posting doesn’t say. Turns out you need a root of the polynomial
-41165056+6728352
2
x
-211591
4
x
-8118
6
x
-108
8
x
+27
10
x
for a value of 3.94164206276524683119682234641
And here’s the diagram for it.
In[]:=
j=
10
0.221
…
;​​k=
5
-0.0476
…
;​​hex=CirclePoints[6];​​verts=TableRotationMatrix[nPi/3].
-199+86k-52
2
k
+8
3
k
4(13-4k+4
2
k
)
,-j-
3
(21+38k-4
2
k
+8
3
k
)
4(13-4k+4
2
k
)
,{n,0,6};​​hexcenters=-9-2k,3
3
-4j-2
3
k,-11+2k,-
3
-4j-2
3
k,-3-2
3
j-2k,5
3
+2j-2
3
k,{-4,-4j},-6+4k,-4
3
-4j,4-2
3
j-4k,6
3
+2j,2-2
3
j,2
3
+2j,2+2
3
j,-2
3
+2j,2
3
j+4k,-6
3
+2j,9-2
3
j-2k,3
3
+2j+2
3
k,9+2
3
j-2k,-
3
+2j+2
3
k,7+2
3
j+2k,-5
3
+2j+2
3
k4;​​Graphics[{EdgeForm[Black],Blue,Polygon[hex+Table[#,{6}]]&/@hexcenters,Red,Line[verts]}]
Out[]=

CITE THIS NOTEBOOK

AI agent AlphaEvolve finds new solution for a packing problem​
by Ed Pegg​
Wolfram Community, STAFF PICKS, May 23, 2025
​https://community.wolfram.com/groups/-/m/t/3466694