Homebrew
Ritual caster. Abacus can cast specific spells as rituals, these spells do not expend a spell slot but the casting takes an extra 10 minutes.
Arcane Grimoire. A spellcasting focus for wizard spells, and Abacus gains a +1 bonus to spell attack rolls and to the saving throw DCs of your wizard spells. He can use this book as a spellbook.
Freezing wave. Melee or Ranged Attack Roll: +6, reach 5 ft. or range 120 ft. Hit: 14 2d10+3 Cold damage. The speed of the creature is reduced by 10ft untill the end of it's next turn.
Spellcasting. Abacus casts one of the following spells, using Intelligence as the spellcasting ability (spell save DC 14, +6 to hit with spell attacks, Arcane Grimoire included):
At will: Mage Hand, Light, Minor Illusion, Prestidigitation, Detect magic (R), Comprehend Languages (R)
2/day each: Mage Armor (included in AC), Earthbind, Sleep, Jump, Identify
1/day each: Guiding Hand, Slow, Erupting Earth, Icingdeath's Frost, Vortex Warp
Tactical order. Abacus can order an allied creature he can see within 30ft, to instantly move up to half the creature's speed into a position Abacus designates. The ordered creture can refuse to move.
Misty Step (2/Day). The mage casts Misty Step, using the same spellcasting ability as Spellcasting.
Protective Magic (2/Day). The mage casts Counterspell or Shield in response to the spell's trigger, using the same spellcasting ability as Spellcasting.