pdelong42 / tumbler-puzzle Goto Github PK

This is going to be my brute-force attempt at finding the solutions to my a puzzle I acquired a number of years ago (I think I may have inherited it from a roommate). Look in the docs directory for the photos of the puzzle at a number of angles, which captures all sides of it. My first task will be to convert this to a convenient schematic, data structure for manipulating.

In short (as you can probably see), the goal is to arrange the tumblers so that all four simple arithmetic equations are true simultaneously. The tumblers can be pulled apart and re-assembled, and rotated, into many permutations. There are some restrictions and simplifying constraints:

• it makes no sense to have two operators adjacent

• by elimination, this means all digits must be singletons (no multi-digit numbers)

• the equal sign should not count as a real tumbler, otherwise it will result in duplicate solutions (x4), which are only different by an offset

• the first tumbler in any permutation should be held stationary while the others are rotated, to avoid further duplicate solutions (again, x4)

To enumerate the solutions:

• there are 24 (4!) permutations of the operator tumblers
• there are 120 (5!) permutations of the digit tumblers
• there are 2880 (24*120) meaningful orderings of all tumblers
• there are 16384 (4^7) possible arrangements of each tumbler ordering (not 4^8, since we always hold one tumbler fixed)
• there are 47185920 (2880*16384) total solutions to try

There are probably some group-theoretic arguments that can be made to really simplify the solution space, but I'm only a group theory novice.

A Lisp, or some other language with lazy evaluation (I plan to use Clojure), is probably the best tool for working through all the permutations and trying each one in-turn. It might help reduce memory footprint (we'll see).

What remains open to interpretation is whether to treat the operations as floating-point, or to treat them strictly as operating on integers. Another is whether to enforce precedence of operators, or apply them naively from left to right.

As a first pass, I'm going to assume left-to-right (for each side of the equation), and take advantage of Clojure's default representation of numbers as rationals. I could always change it later.