Can You Play 3DS Games on a Nintendo DSi? No — Hardware Incompatibility Explained

No. The Nintendo DSi cannot play Nintendo 3DS games. The hardware, media, and software are fundamentally incompatible.

Key Technical Barriers

Processor & GPU: 3DS has a dual-core ARM11 + dedicated GPU; DSi uses a single ARM9 + ARM7. 3DS is ~10x more powerful.
Game Cards: 3DS cards are physically different (tab on side) and use a new interface. DSi slot cannot read them.
Operating System: 3DS runs a modern OS with multitasking, StreetPass, and encryption. DSi firmware cannot boot 3DS software.
Stereoscopic 3D: 3DS games are built for the parallax barrier screen; DSi has no 3D display.

Flashcarts (R4, etc.): Only run DS/homebrew code. None can execute 3DS ROMs on DSi hardware.
Emulation: DSi is far too weak to emulate 3DS. Even New 3DS struggles with some titles.

Yes, 3DS plays DS/DSi games. The 3DS family (3DS, 3DS XL, 2DS, New 3DS) is backward compatible with nearly all DS and DSiWare titles.

Console	Plays DS Games?	Plays DSi Games?	Plays 3DS Games?
DS / DS Lite	Yes	No (DSi-enhanced only)	No
DSi / DSi XL	Yes	Yes	No
3DS Family	Yes	Yes	Yes

To play 3DS games, you need a 3DS family system.

Nintendo DS