Not gonna get a chance to reply tomorrow, so here it is:
First off, it helps to think in terms of "pixels per inch" (ppi), and not "dots per inch".
38 cm / 2.54 = 14.96 inches
14.96 inches * 300 pixels per inch (output) = 4488 pixels of image data
Since a 6x6 frame is really about 56mm x 56mm, then:
56mm / 25.4 = 2.20 inches
4488 pixels / 2.20 inches = 2040 pixels per inch.
In order to get maximum image quality for your 38cm x 38cm print, you should try to scan your 6x6 frames at a resolution equal to or greater than 2040 ppi.
A similar calculation can be done for your 50cm print, yielding a scan resolution of approximately 2680 ppi.
Now, for cases in which the viewing distance is greater than approximately arm's length, the 300 ppi requirement can be relaxed somewhat. A 50cm x 50cm print may appear acceptably sharp from 6 feet viewing distance even when printed at 200 ppi output resolution, for example. But that's a topic for some other time.
--Greg