Snake mistake retake!
Sep. 30th, 2024 02:37 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo! I haven't posted here in a while. Several things I mean to post but I've been busy with other stuff. And most recently I've been busy with a return to the Snake problem!
So, at the suggestion ofjoshuazelinsky, I submitted the Snake problem as a second-year research problem to PROMYS. And the students (unfortunately I don't know their names, aside from one Rakesh Rivera) found that I made some incorrect claims! Here's a big one -- I said that Θ2,2,k was winnable, but in fact it's only winnable for k=1 and k=2!
So, yeah, I had to reevaluate some things. Was the family that Justin found actually winnable? Yes, it was. OK, but then how does one minimize it? (On here and on the webpage I only described the minimized version, but that's not how he described it to me; see the webpage, which now describes the non-minimized version.)
I haven't thoroughly checked it, but I think I have a corrected answer. My previous answer (again, see the webpage) was correct... when k≤m. For m>k, it's a bit more complicated. This is why Θ2,2,1 and Θ2,2,2 are winnable, but Θ2,2,k is not for k≥3. (In fact, Justin's graph is already minimal when m=2 and k≥3.)
But like I said, I think I've fixed it now!
Now to update more stuff. I'll be back with more posts later...
So, at the suggestion of
joshuazelinsky, I submitted the Snake problem as a second-year research problem to PROMYS. And the students (unfortunately I don't know their names, aside from one Rakesh Rivera) found that I made some incorrect claims! Here's a big one -- I said that Θ2,2,k was winnable, but in fact it's only winnable for k=1 and k=2!So, yeah, I had to reevaluate some things. Was the family that Justin found actually winnable? Yes, it was. OK, but then how does one minimize it? (On here and on the webpage I only described the minimized version, but that's not how he described it to me; see the webpage, which now describes the non-minimized version.)
I haven't thoroughly checked it, but I think I have a corrected answer. My previous answer (again, see the webpage) was correct... when k≤m. For m>k, it's a bit more complicated. This is why Θ2,2,1 and Θ2,2,2 are winnable, but Θ2,2,k is not for k≥3. (In fact, Justin's graph is already minimal when m=2 and k≥3.)
But like I said, I think I've fixed it now!
Now to update more stuff. I'll be back with more posts later...
