Due nature of the static websites it is not possible to achieve in the output HTML. The only solution can be creating JSON object with posts to load and then display them randomly, but as I suppose it won’t work for the search engine crawlers as it will be a content dynamically generated by JS script.
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In my case I don’t need random posts on every reload. I want random posts on export/update of the web page. Ideally, I could randomize the array with a helper function. It would be even better if there was a parameter for this in Publii. Is basically only 10 lines of code 😉