First post to Math Stackexchange. Recently I was posed with a mathematically intensive computer programming challenge that I am completely at a loss for solving. To give you some info on my math background, I'm a senior physics major and computer science minor at university and have recently completed a course in linear algebra in addition to having taken calc 1, 2, multivariate calc, and diff eq. I have some small experience with discrete mathematics and number theory from an intro computer science course. I have applied linear algebra, discrete mathematics, and algebraic coding courses scheduled in the next 6 months but am not sure if I will cover the material to answer my problem. I would wait until after taking the courses to see if I can apply anything new to it but this problem has been driving me nuts.
I'm not necessarily looking for a solution but rather some areas to look into that may help me to solve it. If more info or explanation is needed, please let me know. The problem is as follows:
From the scans of the nebula, you have found that it is very flat and distributed in distinct patches, so you can model it as a 2D grid. You find that the current existence of gas in a cell of the grid is determined exactly by its 4 nearby cells, specifically, (1) that cell, (2) the cell below it, (3) the cell to the right of it, and (4) the cell below and to the right of it. If, in the current state, exactly 1 of those 4 cells in the 2x2 block has gas, then it will also have gas in the next state. Otherwise, the cell will be empty in the next state.
For example, let's say the previous state of the grid (p) was:
.O.. ..O. ...O O...
To see how this grid will change to become the current grid (c) over the next time step, consider the 2x2 blocks of cells around each cell. Of the 2x2 block of [p, p, p, p], only p has gas in it, which means this 2x2 block would become cell c with gas in the next time step:
.O -> O ..
Likewise, in the next 2x2 block to the right consisting of [p, p, p, p], two of the containing cells have gas, so in the next state of the grid, c will NOT have gas:
O. -> . .O
Following this pattern to its conclusion, from the previous state p, the current state of the grid c will be:
O.O .O. O.O
Note that the resulting output will have 1 fewer row and column, since the bottom and rightmost cells do not have a cell below and to the right of them, respectively.
Write a function
gis an array of array of bools saying whether there is gas in each cell (the current scan of the nebula), and return an int with the number of possible previous states that could have resulted in that grid after 1 time step. For instance, if the function were given the current state
cabove, it would deduce that the possible previous states were
p(given above) as well as its horizontal and vertical reflections, and would return 4. The width of the grid will be between 3 and 50 inclusive, and the height of the grid will be between 3 and 9 inclusive. The answer will always be less than one billion (10^9).
Inputs: (boolean) g = [ [true, false, true], [false, true, false], [true, false, true] ] Output: (int) 4 Inputs: (boolean) g = [ [true, false, true, false, false, true, true, true], [true, false, true, false, false, false, true, false], [true, true, true, false, false, false, true, false], [true, false, true, false, false, false, true, false], [true, false, true, false, false, true, true, true] ] Output: (int) 254 Inputs: (boolean) g = [ [true, true, false, true, false, true, false, true, true, false], [true, true, false, false, false, false, true, true, true, false], [true, true, false, false, false, false, false, false, false, true], [false, true, false, false, false, false, true, true, false, false] ] Output: (int) 11567
I have looked into coding theory but don't think this falls into any of those categories. I have explored options in bit masking image processing, and cellular automata. The closest I have come to a solution is in researching Margolus neighborhood 2D cellular automata. As this isn't reversible by definition, I haven't found a solution or even general method to find a solution. I'm just curious where I should look. I know this has to have an answer stemming from a field in mathematics but it doesn't resemble anything I currently know.