Elastic Block

We know that, $\text{Force} = \text{Pressure } \times \text{Area}$ In all cases Karen only compresses the block. So she exerts equal pressure $P$ in any case. The only factor which now decides is Area $A$ .

$\therefore$ Force is directly proportional to the Area of the surface. Less the surface area less is the force exerted . In all the given cases less area will be in the case of $3 \times 4$ so, less force is exerted in this case.

The solution is quite simple ( when you've found it :) ). First we have 3 possible areas for the 6 faces of this block :

 - 3 * 4 = 12  square units

 - 3 * 5 = 15 square units

 - 4 * 5 = 20 square units

Now, imagine that we have a force (in Newtons). When we put a force on a surface, the force is concentrated where the contact point is (here, our finger for eg) but there is a transmission of this force through the entire surface, with lower amount of effects than the contact point.

Then, lower the surface is, lower is the force transmission so that the force could be considered as 'more concentrated' so that you need a lower force to do something equivalent than a bigger surface.

From a totally mathematical point of view, we can say this if we consider y our force in Newtons : $\frac{y}{12}$ >= $\frac{y}{15}$ >= $\frac{y}{20}$

As I said before, lower is the surface, the less the force is "dispatched"; with this, we can conclude that the solution is 3 * 4 because it is the smallest area of this block.