Two Equals Four 

Because the exponential tower is infinite, we can also write it as $2=x^{\(x^x^.^.^.\)}.$ But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have $2=x^2.$
Hence $x=\sqrt{2},$ and so $\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^.=2.$
Now consider the equation $4=x^x^x^.^.^.$ and again let's solve for x.
As before, we can write it as $4=x^{\(x^x^.^.^.\)},$ and again, the part in brackets is the same as the whole, and so now we get $4=x^4.$
But take the square root of each side and we get $2=x^2,$ and so again $x=\sqrt{2}.$ So now we have $\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^.=4.$
So $\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^.$ is 2, and it's also 4.
Hence 2=4 (and halving it means 1=2).