**Collatz's Staircase**

*Wanted to entertain the idea of a mathematical SCP. This is from an actual bad dream.*

A structure that appears to be a four-storey apartment building from the exterior. Entering through the lobby, entrances to two staircases can be seen, one on each side of the building. The complex attracted Foundation attention after reports of people disappearing into it.

Walking upwards on the staircase on the East side of the building takes you up one storey as expected. However, the staircase extends past the four storeys as seen from outside the building, and continually walking upwards takes one through a series of storeys that contain similar, empty apartment suites. Plaques installed on each floor confirm that the floors are indeed distinct: the floors are labelled, as expected, with successive natural numbers.

It is impossible to walk downwards on the staircase on the East: one may walk up and down on a flight of stairs at will until one reaches the next floor, at which point the staircase behind is rendered inaccessible (replaced by a wall once you look away? Not sure how to execute this.)

Walking across the complex at any given floor takes one to the West side staircase on the same floor. The West side staircase is similar in appearance to the East side staircase. It is always possible to move in either direction on the West side staircase. However, instead of taking one from the nth story to the n+1th, walking one floor upwards on the West side staircase staring from floor n instead takes one to floor 3n+1 (as evident from the labelling of the plaques). Walking one floor downwards from floor n takes one to floor n/2. (In this fashion, starting from an arbitrary floor, oftentimes hundreds of inter-floor trips would be required to return one to the first floor. See the Collatz conjecture) This is true even for odd n: the labelling on the plaque at the destination accordingly designates the floor as a fraction.

It is apparent that, once one arrives at a non-integer floor, it is impossible to return again to the integer-numbered floors. Tests with remote controlled robots / D-class personnel have found no other methods to leave the building once a non-integer floor has been entered, and invariably end with contact being lost as communication devices carried by test subjects run out of power.

The difficulties I'm having with this one are mostly related to the fact that it's relatively hard to naturally explore through a narrative; despite the entire analogy to the mathematical object was conceived very much at once.