This is a good starting point to discuss where the CLT breaks down: a log-normal doesn’t look all that normal, but it’s normal enough that it will converge under CLT. So at what point does CLT break down? Well, the log-normal tail is fat but not fat enough to do it; you need to go to something without finite variance, like a Cauchy distribution… You could also discuss the log-normal variation of the CLT: why does it not happen for your chess match length when the chess matches are log-normal, and what sort of scenario might make it do so?
This is a good starting point to discuss where the CLT breaks down: a log-normal doesn’t look all that normal, but it’s normal enough that it will converge under CLT. So at what point does CLT break down? Well, the log-normal tail is fat but not fat enough to do it; you need to go to something without finite variance, like a Cauchy distribution… You could also discuss the log-normal variation of the CLT: why does it not happen for your chess match length when the chess matches are log-normal, and what sort of scenario might make it do so?