If I continue to work on my game in construct 3 beta version r121 I notice I can no longer open it in the current version r120. If I continue to work on it in the beta (r121) will it still work when r121 is made in to the main version? Or is that a bad idea because it might not work?
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Releases never have the same number, so if a beta is r120 and they release a stable it will be r121. The next beta would be r122. Releases are always incremental regardless of if that release is deemed stable or beta.
You can open you project in any version, beta or stable, the same or later as the release it was saved in but not prior to that, as a later version might have code changed or might support new features which are not in the earlier releases.
Keep in mind that beta is beta for a reason and that if you are worried a new version will break your game you should stick to stable builds, or at least always have backups.