Is it true that cities only use resources within 3 tiles from your city, no matter your city borders?
A friend claimed this and I can neither confirm nor refute it so far. Is this true?
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No, it is not true.
While a city can only work a tile within three steps, it will still provide access to a strategic or luxury resource for your empire if the relevant improvement is built upon that tile and the tile is within the city boundary.
It is true, yes. You can confirm this in the civilization wikia:
To have access to any Resources, especially Strategic and Luxury ones, you need them to be inside your territory. Founding cities near them (1 to 3 tiles away) is the best way to assure that. Never forget that a city may only work tiles that are at most 3 tiles away from its main city tile. So, when choosing the spot, try to include as many Resources within 3 tiles of it as you can. Even though your budding city won’t include them into its territory yet, it will start growing it later with exactly this in mind.