How about an example of where the numerical value of $\pi$ is what makes a problem not solved? The moving sofa problem. See Dan Romik's paper.

An original upper bound on the moving sofa constant given by Hammersley is $2\sqrt{2}\approx 2.828$ which has been improved to $2.37$ by Romik.

There is an example given below of a sofa with area $\pi/2+2/\pi\approx 2.2074$. Note that this is not a solution because $\pi/2+2/\pi<2.37.$

Actually, $x/2+2/x=2.37$ has two solutions, $x\approx 1.09843,3.64157.$ In particular for $x\in (1.09843,3.64157)$ we have that $x/2+2/x<2.37$.

The example given below is not optimal because $\pi\in(1.09843,3.64157)$.

I believe this qualifies because this shape is a natural attempt and one of the first things one would try. The problem is interesting and nontrivial precisely because one of the first things one would try doesn't work. The $\pi$ shows up due to circular shape and hence isn't an arbitrary choice.

How about an example of where the numerical value of $\pi$ is what makes a problem not solved? The moving sofa problem. See Dan Romik's paper.

An original upper bound on the moving sofa constant given by Hammersley is $2\sqrt{2}\approx 2.828$ which has been improved to $2.37$ by Kallus and Romik.

There is an example given below of a sofa with area $\pi/2+2/\pi\approx 2.2074$. Note that this is not a solution because $\pi/2+2/\pi<2.37.$

Actually, $x/2+2/x=2.37$ has two solutions, $x\approx 1.09843,3.64157.$ In particular for $x\in (1.09843,3.64157)$ we have that $x/2+2/x<2.37$.

The example given below is not optimal because $\pi\in(1.09843,3.64157)$.

I believe this qualifies because this shape is a natural attempt and one of the first things one would try. The problem is interesting and nontrivial precisely because one of the first things one would try doesn't work. The $\pi$ shows up due to circular shape and hence isn't an arbitrary choice.

How about an example of where the numerical value of $\pi$ is what makes a problem not solved? The moving sofa problem. See Dan Romik's paper.

An original upper bound on the moving sofa constant given by Hammersley is $2\sqrt{2}\approx 2.828$ which has been improved to $2.37$ by Romik.

There is an example given below of a sofa with area $\pi/2+2/\pi\approx 2.2074$. Note that this is not a solution because $\pi/2+2/\pi<2.37.$

Actually, $x/2+2/x=2.37$ has two solutions, $x\approx 1.09843,3.64157.$ In particular for $x\in (1.09843,3.64157)$ we have that $x/2+2/x<2.37$.

The example given below is not optimal because $\pi\in(1.09843,3.64157)$.

How about an example of where the numerical value of $\pi$ is what makes a problem not solved? The moving sofa problem. See Dan Romik's paper.

An original upper bound on the moving sofa constant given by Hammersley is $2\sqrt{2}\approx 2.828$ which has been improved to $2.37$ by Romik.

There is an example given below of a sofa with area $\pi/2+2/\pi\approx 2.2074$. Note that this is not a solution because $\pi/2+2/\pi<2.37.$

Actually, $x/2+2/x=2.37$ has two solutions, $x\approx 1.09843,3.64157.$ In particular for $x\in (1.09843,3.64157)$ we have that $x/2+2/x<2.37$.

The example given below is not optimal because $\pi\in(1.09843,3.64157)$.

I believe this qualifies because this shape is a natural attempt and one of the first things one would try. The problem is interesting and nontrivial precisely because one of the first things one would try doesn't work. The $\pi$ shows up due to circular shape and hence isn't an arbitrary choice.