I hope you enjoyed Saturday’s Mac Riddles, episode 370. Here are my solutions to them.
1: Focussed lamp for alphabetical list to find files again.
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Spotlight index
Focussed lamp (a spotlight) for alphabetical list (an index) to find files again (what rebuilding it is claimed to achieve).
2: Fix authorisations still offered in recovery, if you dare.
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Permissions
Fix authorisations (permissions) still offered in recovery (it’s still available in Recovery mode), if you dare (just don’t try using it!).

3: Working area rebuilt to restore icons.
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Desktop
Working area (desktop) rebuilt (to repair it) to restore icons (rebuilding it in Classic Mac OS restored associations between apps and document icons).

The common factor
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Rebuilding, repairing or resetting these have been popular panaceas for fixing problems.
I look forward to your putting alternative cases.
