I recently came up with a series of mathematical puzzles based loosely on Rubik's puzzles(or should I say combination puzzles). They go like this: you have an n x n group of numbers where you can shift either a row or a column in any cardinal direction(left,right,up, or down) however you must increment every number involved by one. I am simply calling these Curtesian Objects and they are solved when all of the elements equal the same number. Curtesian Objects can also be described as tuples ex. x = 2x2 (1,2,3,4) where the (1,2) and (3,4) are the rows of x and (1,3) and (2,4) are the columns. An example of shifting would be shifting the top row of x left making x = (3,2,3,4). Keep in mind it only looks like switching because both n's are 2.
Hopefully I explained Curtesian Objects well enough and if so lets describe my problem. A Curtesian Object is solvable if all of its elements can become the same. An example is x (1,2,3,4) which is solvable to 5 and the steps are:
I omitted the directions due to it being obsolete when n = 2
- Left Column Shift(or 1L for 1 shift) to (4,2,2,4)
- Bottom Row Shift(or 1B) to (4,2,5,3)
- Double Right Column Shift(or 2R and adds 2 to both elements) to (4,4,5,5)
- Top Row Shift (or 1T) to (5,5,5,5)
The puzzle is how can you tell if a Curtesian Object is solvable and if so to what number? I've had some success using algebra since it helps in finding patterns of solvable puzzles but not in solving puzzles that I can't prove are unsolvable. Also looking into how changing n and the number of dimensions changes the problem would prove interesting. As a side question what do you call a problem that is hard to solve and hard to verify?