Unlikely. Halo: CE (1) and Halo 2 were on the original Xbox, Halo's 3 and 4 are on the 360, so 5 and 6 will most likely be on the Xbox One.
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As far as I know, you can play all original XBOX games on the XBOX 360. It may be different for some of the first XBOX games but both Halo and Halo 2 are playable on the XBOX 360.
the xbox 360
Halo 4
Yes, you can since I am a big fan of Halo myself I have tried it.
If you are interested in purchasing an XBox 360 Halo game, then you will find several online retailers available. Walmart, eBay, Target, and Gamestop are just a few of the retailers that offer the XBox 360 Halo game.