Newton's 3rd Law: How can I break things? If I punch a wooden board hard enough and it breaks in two, has the board still exerted a force of equal magnitude on my fist? 
When the board breaks in two due to my force, the halves have a component of acceleration in the direction of my striking fist...that implies the board did not exert an equal and opposite reaction, no?
 A: The board did exert an equal and opposite force, but your mass is considerably greater than the mass of the board so all the friction between you and the ground keeps you from accelerating.
If you punched the board while standing on perfectly slippery ice, or in a vacuum, you would accelerate also, but at a much smaller rate than the board due to the mass difference.
If you could ignore all losses (friction, the board breaking, etc.) then all Newton's 3rd Law really is saying is that the center of mass of the system doesn't change when you punch the board. So the board accelerates in the direction of your punch and you move away from it at rates that keep the system center of mass constant.
A: 
If I punch a wooden board hard enough and it breaks in two, has the board still exerted a force of equal magnitude on my fist? 

Yes.

When the board breaks in two due to my force, the halves have a component of acceleration in the direction of my striking fist...that implies the board did not exert an equal and opposite reaction, no?

Incorrect, the board did exert an opposite reaction - where do you think your bruised knuckles will come from? The transfer of momentum from your body to the board requires an opposite force to decelerate your fist or you could punch through arbitrarily thick material.
That the opposite force will be equal in magnitude is a consequence of conversation of momentum applied to a 2-body interaction - modern physics no longer postulates the 3rd law.
A: 
If I punch a wooden board hard enough and it breaks in two, has the board still exerted a force of equal magnitude on my fist? 

Good question but short answer: Yes. Exactly..!
The Newton's third law (a contact force) applies to every-day life like Jumping on the floor, Walking on slippery ground, balloons etc. So, the board has also exerted an equal force on your fists...
How is this? (similar to $\vec{F_{AB}}=-\vec{F_{BA}}$) Whenever you exert a force on an object, the object also exerts an equal normal force on you so that both cancels out. Without it, your fist won't stop accelerating and you'd probably pierce through the ground. One thing to be noted is that these action-reaction pairs always act on two different bodies and depends on their masses. Here, the board does not have enough mass to withstand your force and hence the break-through. It could be seen from this example.
As a ball falls on earth, you could tell that the ball is pulled by earth or according to 3rd law, earth is pulled by ball). Amazed of it, Eh..?
Take the mass of ball to be $10kg$, then force exerted by Earth on ball is $F=mg=98N$. According to the 3rd law, this force is also exerted by the ball on earth. Hence, acceleration of Earth towards ball is $a=F/m=\frac{98}{5.98×10^{24}}=16.38×10^{-24}ms^{-2}$ which is too small to be measured. There are greater probabilities for the forces to be same, but not necessarily their acceleration..!
A simulation for Newton's cradle which is also an example of the 3rd law...

When the board breaks in two due to my force, the halves have a component of acceleration in the direction of my striking fist...that implies the board did not exert an equal and opposite reaction, no?

NO...
Actually the Newton's 3rd law is stated as:


*

*All forces result from interactions between pairs of objects, each object exerting a force on the other. The two resultant forces have same strength but in exactly opposite directions.


It seems that the breaking of board has confused you. The board-breaking is due to a Stress-Strain mechanism which is not necessary here..! It depends on the type of material, magnitude of force exerted on it, etc. When you apply the force on the board, the work done  by you is stored as potential energy in the board. When it exerts the equal force on you, it also experiences the same force and when this force exceeds the breaking stress, the board BREAKS..! The potential energy is then dissipated in the form of heat.
My thought is that, Simultaneous action-reaction pairs are observed both on board and on your fist. But, the board can't withstand the action-reaction due to it's low mass and small breaking stress..!
