Two Equals Four

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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).