You have been abducted by a sadistic math wizard (don’t you hate it when that happens?). He ushers you into a plain but cozy-looking room, with a hardwood floor, a few exotic-looking rugs, and wood paneling on the walls.
He hands you a small polished metal cube. “Inside this cube”, he explains, “is a positive rational number. If you can correctly tell me its numerator and denominator in lowest terms, I will let you go free. If you guess wrong, though…” He pauses to indulge in an evil chuckle. “Let’s just say that my sharks are hoping you will fail.” He strides out and slams the door behind him.
Of course you try to open the cube, but you can’t find any way to do it: you don’t see any hinges, or a lid, or even any kind of seam at all, just smooth metal. You try throwing it on the floor to see if it will shatter, but you only succeed in denting the floor. You next try the door, but of course it is locked, and seems extremely solid.
You sigh and start to look around the room. There are no windows, but you quickly notice that a few of the wooden panels have various holes, knobs, and inscriptions on them. You go to examine them more closely.
On the west wall is a panel labelled “RECIPROCATOR”. To one side there is a square hole labelled with the letter ; next to it is another hole labelled . The holes look like they might be just about the same size as the metal cube, so you hold your cube up next to the hole labelled to compare their sizes. Suddenly, with a whirring sound, something starts to pull the cube out of your hand, into the hole; you desperately try to hang on, but the mechanism is too strong. You experience a brief moment of panic as the cube slips from your fingers, but instead of disappearing, it stops as soon as it is flush with the panel. There are more whirring sounds from inside the wall, and a few moments later, your cube slowly emerges from the hole again, along with new cube from the hole labelled . They look identical; you are extremely glad at this moment of the decision you made, eleven years ago, to carry a permanent marker in your pocket at all times Just In Case. You carefully label the original cube and the new cube so you will not get them mixed up, since you are starting to get the feeling that you might end up with a lot of these.
On the north wall is a panel labelled “ARITHMETIZER”. There are three square holes: one labelled , one labelled , and one labelled . However, you notice a little knob below the symbol; by turning it you are able to change the into or . You turn it back to and, sure enough, when you put your two cubes next to the holes labelled and , they are sucked into the machine briefly, and after some whirring sounds they spit back out, along with a third cube (which you carefully label) ejected from the third hole.
The east wall has no panels, but it does have a window with a stunning view of the forest you were hiking in just a few hours ago. You realize, belatedly, that the stories that crazy old man at the lodge told about the “haunted math forest” may have had some truth to them after all.
On the south wall is a panel labelled “COMPARATOR”, with two holes labelled and . To the right of those holes, under a blank space on the panel, is a label that reads . You try putting in your original cube and the one that came out of the ARITHMETIZER. After the usual whirring, part of the panel over the label slides open for a few seconds, revealing a giant letter T. You swap the two cubes and put them back in, and sure enough, this time the panel slides open to reveal a giant letter F.
In the middle of the room is a small table with a chair and a lamp. On the desk, it looks like the wizard has thoughtfully provided you with a quill, a pot of ink, and a stack of blank paper. You have the first few ticklings of some ideas, so you sit down at the desk, look thoughtfully at the ceiling for a few seconds, then begin to write.
Can you escape? Or will you be feeding the sharks? What’s your best strategy?
Steel cube photograph by Zai Divecha