[Harryobr]

My have office chair have six. And what?

[jip24]

Imagine placing an elastic band around all of the wheels together - so a square if there are four wheels, a pentagon if there are five. To stay stable, the centre of mass needs to remain over the bounded area.

Suppose one wheel breaks - the elastic band pulls inwards accordingly, held by the others. Assuming the centre of mass is in the middle of the chair, are you still stable? If you start with four wheels, and end up with three, the elastic band makes a right angled triangle, with the long edge passing right through the centre of mass....so any slight wobble in the wrong direction and you'll fall over. With five wheels, if one breaks, you're still stable.

[Translunary]

Most of the executive office chairs have castor wheel having a five-star base to give you comfort and balance. Having these wheels you can move around the place in 360-degree rotation easily.

[Hayseed]

Five spokes narrow the tilting plane(area).

[danfangaoqingmv]

The office chairs are the more stable the wheels are? Like SPAM office chair.

[Georgesmith12]

I think number of wheels depend on the chair's design.

If the chair's design can't evenly distribute the weight equally on 4 wheels to balance your posture and position. Then the manufacturer increases the number of wheels. And in case of 5 wheels, the chairs are designed in a way that it must be having 5 wheel otherwise, will can cause issue to you. 

[Jeffery21]

So you are mobile and can easily move between/across desks/sections of your cubicle. Having a four legged chair in an office environment would be horrendous.

[khan648]

Office chair casters are important for the chair's performance and your comfort. Choose the right type of caster wheels for your chair. Desk chair caster wheel options include:

-Tired rubber wheels for carpet or hardwood floors.

-Smooth plastic wheels for hardwood floors.

-Carpet caster cups to protect your flooring.

MOD EDIT: SPAM LINK REMOVED