You are on a frictionless horizontal plane. How can you get off if no horizontal force is exerted by pushing against the surface

The correct answer is By splitting or sneezing.

To understand how one can move on a completely frictionless horizontal plane, we must look at the laws of motion, specifically Newton's Third Law of Motion (every action has an equal and opposite reaction) and the Conservation of Linear Momentum.

Normally, when we walk, run, or jump, we rely on the force of friction. We push backward against the ground with our feet, and the frictional force exerted by the ground pushes us forward. However, on a frictionless plane, the surface cannot exert any horizontal force on us. Therefore, actions like running, jumping vertically, or rolling that rely on traction or horizontal surface reaction forces are impossible or cannot generate horizontal motion.

To move horizontally without external friction, we must use the principle of conservation of momentum. Since the system (our body and any ejected matter) starts at rest, its initial momentum is zero:
$P_{\text{initial}}=0$
If we spit, sneeze, or throw an object (like a coin or a piece of clothing) in one direction, that ejected mass carries momentum in that direction. Let the mass of the ejected matter be $m$ and its velocity be $v$. Let the mass of our body be $M$ and our resulting velocity be $V$. By conservation of momentum:
$MV+mv=0$
Solving for our velocity:
$V=-\frac{mv}{M}$
The negative sign indicates that our body will slide in the opposite direction of the spit or sneeze. Thus, ejecting mass (by spitting or sneezing) generates a recoil force that propels us across the frictionless surface, allowing us to get off the plane.