Who listens to their architect anyway?

Now suppose the lamp has a radius $r$ and is suspended a height $h$ off the ground in a room with height $2h$. Again, the radius of the shadow on the ceiling is $R.$

For whatever reason, the restaurant’s architect insists that she wants $r$, $h$, and $R$, as measured in feet, to all be whole numbers. What is the smallest value of $R$ for which this is possible?

Let's imagine that this restaurant is named, say Cafe Pythagoras. Abstracting the Classic problem's solution by having a circle of radius $r$ centered at $(0,h),$ then we have the minimal distance from the lamp to the reflected path of the light from the ground to the ceiling as $d^* = 2h \sin \theta.$ Therefore, we see that $$R = \min \{ 3h \tan \theta \mid 2h \sin \theta \geq r \} = 3h \tan \left( \sin^{-1} \left(\frac{r}{2h} \right) \right) = \frac{3hr}{\sqrt{4h^2 - r^2}}.$$

Since the architect at Cafe Pythagoras wants to make sure that $r$, $h$ and $R$ are all integers, we should start searching through Pythagorean triples for possible solutions, since if the denominator $\sqrt{4h^2 -r^2}$ is not an integer, then there is no way for $R$ to be an integer. In particular we need $2h$ to be the hypotenuse of a Pythagorean triple and either $r=2a$ or $r=2b$. Naturally, let's start with the triple $(3,4,5)$. So we set $h=5$ and try $r=6,$ which yields the undesirable $R = \frac{45}{4} \not\in \mathbb{N}.$ If instead, $h=5$ and $r=8$, then we get $R = 20,$ so at least we know that this your architect is not asking for the impossible. By taking scaled multiples, say $h=5k$, $r=8k$ and $R=20k,$ for any $k \in \mathbb{N},$ we see that in fact the architect can have infinitely many solutions, with the smallest one generated by the (3,4,5) primitive Pythagorean triple having a shadow radius of $R=20$.

Let's assume that we have some integers $a, b, c \in \mathbb{N}$ with $a^2 + b^2 = c^2,$ $\gcd (a,b,c) = 1,$ and $c \gt 5$ and see if we can come up with any other solutions for our Pythagorean architect. Without loss of generality, let's see what would happen if we set $h = c$ and $r = 2b.$ In this case we get $$R = \frac{6bc}{\sqrt{4h^2 - (2b)^2}} = \frac{6bc}{2a} = \frac{3bc}{a}.$$ Since $\gcd(a,b,c) = 1,$ if $a \ne 3,$ then $R \not\in \mathbb{N}.$ Since the only primitive Pythagorean theorem that contains $3$ is $(3,4,5),$ we see that if $(a,b,c) \ne (3,4,5),$ then $R \not\in\mathbb{N}.$

Therefore, we have confirmed two very important things: Firstly, that the smallest possible value of $R$ at Cafe Pythagoras is $R=20,$ when $h=5$ and $r=8.$ And, lastly, but perhaps equally important, I don't think that I'll be going to Cafe Pythagoras. With a sizeable 8' radius orb floating with its center only 5' off the ground, that means that the orb would be larger than the actual ceiling height???!!! The beam of light does escape the ginormous lamp and properly reflect off the floor and up to the ceiling after 20' as promised, but something seems off about the architectural demands. Perhaps I'm just too old to be the target audience for this restaurant. I hear the food is good though!