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How detailed do the design logs need to be?
For example, if we have a piece of cardboard, do we need to indicate its dimensions?
What about something like legos?

For the design log, you need to include all the materials you have used, (4.b.1). However, I do not believe you have to be extremely detailed, and something like a URL to the website you bought that object from is sufficient, or in the case of more general items like your cardboard, stating that you used cardboard should work. (unless its a special type of cardboard, like corrugated cardboard.)

This varies a lot by whim of the event supervisor.

At our state tournament, one ES was merrily penalizing all comers for not specifying the type of adhesive used in the list of materials.

I would expect a materials list to include dimensions for each piece. For legos, I'd probably specify their dimensions in studs, rather than in mm, although I might well give both sets of dimensions.