Edited by Richard Walker, Sunday 15 February 2026 at 12:47
A few days ago I posted a puzzle I'd seen, to find five circles that would between them pass through all 25 dots in a square 5 x 5 grid. Here is the solution I gave, just found by using trial and error.
That got me interested, so then I tried with a 6 x 6 grid, again using trial and error and exploiting some obvious symmetries and was pleased to find this very nice solution with 6 circles, which has all the eight symmetries of a square.
This got me even more interested, so I did a search to see if I could find any literature on the genera case of circles. I didn't find a lot but I did locate this blog post where the author had written a program to do a brute-force search to the 5 x 5 case and found 84 essentially different configurations that solve the problem.
Then I looked for a trial and error solution to the 7 x 7 case but found it more difficult and didn't make much progress, so I decided to recruit ChatGPT 5.2 as a research collaborator. I first asked if it could locate anything about the general problem but it drew a blank. So then I wrote quite a long and detailed prompt asking it if it could solve the 5 x 5 and 6 x 6 cases.
It quickly wrote and ran a Python program to do a search and after a minute or two came back with the equations of circles solving the 5 x 5 and 6 x 6 cases. I've not yet checked but I think they are essentially the solutions I've shown above. I felt very encouraged so then we moved on to the 7 x 7 case.
At first the best it could do was 9 circles but after tweaking its algorithm it reduced this to 8 and gave me the equations. I typed them into the brilliant and free online application Desmos [2] and here's the result. It works!
Is this the minimum possible? I asked CharGPT and it had a go, writing and running code, but it exhausted its quota of processing cycles without finishing. It then displayed the program and suggested I run it on my local computer, which I did and after a few minutes it finished with the message
'No 7-circle cover exists (CERTIFIED)'
I should say that throughout this conversation ChatGPT had been at pains to stress that I should try to independently verify whatever it told me and of course I had planned to do that anyway. So I next asked, as a test of its algorithm, how many different solutions exist for a 5 x 5 grid. And it said the solution it had given was
Unique
This really surprised me, because I'd found a website that seemed to compute solution to the 5 x 5 and claimed to have found 84. So then I tried to get ChatGPT to re-evauate its conclusion, even sending an image of a different solution and explaining why it was different. But I couldn't shake ChatGPT on this.
So I then I resumed my search for pages that might have relevant content but this time I asked Copilot and it did better that ChatGPT and found a page in the Online Encyclopedia of Integer Sequences. This has lots of information, including the number of circles needed up as far as the 12 x 12 case (which takes 15), the fact that the only cases with unique solutions are 3 x 3 and 8 x 8 (so 5 x 5 is not unique and ChatGPT was indeed wrong about that) and confirmation that 7 x 7 needs 8 circles, as indicated by ChatGPT's program.
And it has a link that would lead to original source of much of this information and would probably answer questions like 'How do we know the 8 x 8 solution is unique? Is by computation or was there some kind of deductive argument?
But unfortunately this link broken and I haven't so far been able the locate the page I want by other routes, so I am paused from the moment.
I started this investigation partly because of the inherent interest of the problem but also because I wanted to explore for myself how useful AIs such as ChatGPT might be as research assistants. In my estimation, very useful indeed, with all the usual caveats about not simply accepting its answers without careful checking. I was very surprised by how well ChatGPT seemed to grasp what I was trying to find out and by its versatility in suggest avenues I might like to have it explore. And I was impressed by how much it got right but also slightly frustrated by its stubbornness when it was wrong.
Stop Press
I just found this page which shows 3 solutions to the 5 x 5 case, and asks 'Are there others?'
Joining Up The Dots
A few days ago I posted a puzzle I'd seen, to find five circles that would between them pass through all 25 dots in a square 5 x 5 grid. Here is the solution I gave, just found by using trial and error.
That got me interested, so then I tried with a 6 x 6 grid, again using trial and error and exploiting some obvious symmetries and was pleased to find this very nice solution with 6 circles, which has all the eight symmetries of a square.
This got me even more interested, so I did a search to see if I could find any literature on the genera case of circles. I didn't find a lot but I did locate this blog post where the author had written a program to do a brute-force search to the 5 x 5 case and found 84 essentially different configurations that solve the problem.
Then I looked for a trial and error solution to the 7 x 7 case but found it more difficult and didn't make much progress, so I decided to recruit ChatGPT 5.2 as a research collaborator. I first asked if it could locate anything about the general problem but it drew a blank. So then I wrote quite a long and detailed prompt asking it if it could solve the 5 x 5 and 6 x 6 cases.
It quickly wrote and ran a Python program to do a search and after a minute or two came back with the equations of circles solving the 5 x 5 and 6 x 6 cases. I've not yet checked but I think they are essentially the solutions I've shown above. I felt very encouraged so then we moved on to the 7 x 7 case.
At first the best it could do was 9 circles but after tweaking its algorithm it reduced this to 8 and gave me the equations. I typed them into the brilliant and free online application Desmos [2] and here's the result. It works!
Is this the minimum possible? I asked CharGPT and it had a go, writing and running code, but it exhausted its quota of processing cycles without finishing. It then displayed the program and suggested I run it on my local computer, which I did and after a few minutes it finished with the message
'No 7-circle cover exists (CERTIFIED)'
I should say that throughout this conversation ChatGPT had been at pains to stress that I should try to independently verify whatever it told me and of course I had planned to do that anyway. So I next asked, as a test of its algorithm, how many different solutions exist for a 5 x 5 grid. And it said the solution it had given was
Unique
This really surprised me, because I'd found a website that seemed to compute solution to the 5 x 5 and claimed to have found 84. So then I tried to get ChatGPT to re-evauate its conclusion, even sending an image of a different solution and explaining why it was different. But I couldn't shake ChatGPT on this.
So I then I resumed my search for pages that might have relevant content but this time I asked Copilot and it did better that ChatGPT and found a page in the Online Encyclopedia of Integer Sequences. This has lots of information, including the number of circles needed up as far as the 12 x 12 case (which takes 15), the fact that the only cases with unique solutions are 3 x 3 and 8 x 8 (so 5 x 5 is not unique and ChatGPT was indeed wrong about that) and confirmation that 7 x 7 needs 8 circles, as indicated by ChatGPT's program.
And it has a link that would lead to original source of much of this information and would probably answer questions like 'How do we know the 8 x 8 solution is unique? Is by computation or was there some kind of deductive argument?
But unfortunately this link broken and I haven't so far been able the locate the page I want by other routes, so I am paused from the moment.
I started this investigation partly because of the inherent interest of the problem but also because I wanted to explore for myself how useful AIs such as ChatGPT might be as research assistants. In my estimation, very useful indeed, with all the usual caveats about not simply accepting its answers without careful checking. I was very surprised by how well ChatGPT seemed to grasp what I was trying to find out and by its versatility in suggest avenues I might like to have it explore. And I was impressed by how much it got right but also slightly frustrated by its stubbornness when it was wrong.
Stop Press
I just found this page which shows 3 solutions to the 5 x 5 case, and asks 'Are there others?'