The electronic structures, band-gap origins and magnetisms of Ti2Cr-based alloys with CuHg2Ti-type structure are studied using the first principles calculations. It is found that Ti2CrK (K=Si, Ge) alloys are semiconductors Ti2CrK (K=Sb, Bi) alloys are predicted to be half-metallic ferrimagnets and their half-metallic band gaps are affected directly by the S states of Sb and Bi atoms. Ti2CrSn alloy is a completely-compensated ferrimagnetic semiconductor. Due to the different band-gap origins of Ti2CrSn alloy in two spin directions, we can adjust the width of band gap by doping engineering. The ferrimagnetic spin-gapless materials are achieved by substituting Si or Ge for Sn. Substituting Fe or Mn for Cr, we gain a series of half-metallic materials. Ti2Cr1-xFexSn and Ti2Cr1-xMnxSn alloys are in ferrimagnetic states. All the half-metallic Ti2Cr-based alloys follow Mtotal=Zt-18 rule (Mtotal is the total magnetic moment and Zt is the valence concentration).