You can for multiplayer games, Poke Transfer, Wi-Fi settings transfer or PictoChat. A 'mini' version of certain games like Super Mario 64 DS can be sent to another DS for demonstration or multiplayer, using DS Download Play.
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Yes you can. You can also connect DS lite to DS and DS Lite to DSi.
in pictochat or download playSimple! Use your DS Download Play!
For the most part, yes you can. If you want to connect with pictochat or DS games (including DS Download), you can connect between a Nintendo DS and DSi. However, for DSi specific games, such as DSi games and certain applications, the Nintendo DS will not be able to connect. But, note that certain DSiWare games that can DS download play still can be connected to the Nintendo DS.
It must be impossible to do this but you can connect a Nintendo DS or DS lite system to a TV
No. You can connect those games to A Wonderful Life/Another Wonderful Life and or Harvest Moon DS/DS Cute. But not to each other.