clm
Posted on: Thu Sep 13, 2012 12:05 am
Posts: 716 Joined: Fri May 13, 2011 6:51 pm

The replacement
The replacement.
Is it possible to obtain 9! (factorial of 9) = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 different Killer Sudokus from a particular one with a unique solution keeping the same structure of cages?.
Is it possible to create a different type of Calcudoku (which we could name “Replaced Puzzles”) similar to the twin puzzles but this time keeping the operators and the structure of the cages though modifying the content (instead of the usual twin puzzles that keep the numbers in their places modifying the cages and the operators)?.
For the purpose of the following hypothesis I define a replacement as the substitution (for instance in a 9x9 Calcudoku) of all numbers 1 thru 9 for another set 1 thru 9 in such a way that there is an straight correspondance one by one between each pair of numbers. Here are some examples:
1 is replaced by a 2 2 >>> 3 3 >>> 4 4 >>> 5 5 >>> 6 6 >>> 7 7 >>> 8 8 >>> 9 9 >>> 1. This first example looks like a “circularization”.
1 is replaced by a 1 2 >>> 2 3 >>> 3 4 >>> 4 5 >>> 8 6 >>> 5 7 >>> 9 8 >>> 7 9 >>> 6. In this case the first four numbers have not been modified, the rest have been replaced “randomly”.
Or a totally “random” replacement:
1 >>> 5 2 >>> 8 3 >>> 6 4 >>> 1 5 >>> 7 6 >>> 2 7 >>> 4 8 >>> 9 9 >>> 3
Let’s suppose we have a totally solved 9x9 Calcudoku (with its unique solution) and that we make a replacement keeping the same cages and operators (we will discuss soon the exceptions to this) but logically recalculating the result of the operations (the result of the cages). I have made several tests with puzzles of different sizes and found that always the solution is unique.
Operations as “x”, “+”, bitwise OR (), mod and exponentation (^) do not represent any problem since always produce a proper result.
The “” operation is not a problem for a 2cell cage but with bigger cages it may be a problem; for instance [9,3,2] (= “4”) with a replacement of [7,8,4] is not possible. In a case like this we need to modify the type of operator to, i.e., “x” or “+”. In some particular cases we could keep the operator “”, for instance, when [9,3,2] is replaced by [5,1,4] (= “0”).
A similar situation appears with the “:” operator, [8,4,1] (= “2:”) can keep the operator for a replacement with [6,2,3] (= “1:”) but it would be impossible for instance for a replacement with [9,5,3] and now we must change the ":" operator, i.e. in this case, to “” (= “1”). If the “:” cage has 2 cells then the replacement by “” is always possible.
In the cases of the “from 0” puzzles the “:” with more than 3 cells have also a problem if two zeros fall in the same cage, [3,3,1] (= “1:”) replaced by [0,0,5] where the “:” operation is impossible (but we could change it to a “5” cage for instance).
Well, the interesting thing is that with those “minor” (and required) changes in the appropriate operators, as far as I have tested, the solution is again unique.
For instance, the 7x7 subtraction only puzzles admit all possible replacements (no problem, as explained, with the 2cell cages of the type “”) so, as there are 7! = 5,040 different assignments (permutations), there are 5,040 puzzles (with unique solution) with the same structure of cages.
It is clear that for a determined Killer Sudoku (all cages are “+”) we can obtain 362,880 different Killer’s with the same structure of cages. This facilitates the programming, designing or analysis since it’s very easy to generate new puzzles (with a unique solution) from any previous one.
A question arises inmediately: Are all of the same level of difficulty?. According to my experience the answer is NO (however this has only been tested manually and not with a sofware). So in the case of the Killer Sudoku, for instance, we could adjust the difficulty level by choosing among those 362,880 replaced puzzles (that is, not all these 362,880 have the same difficulty level). Probably part of the difficulty is due to the structure of cages (and the type of operation in the case of Calcudokus) and part is due to the replacement.
Finally, using the exposed idea, we could create a different type of “double” puzzle, named, i.e., “replaced puzzles” both with the same structure of cages and operators but with a replacement of numbers inside; since both would have a unique solution, we could use the pair of puzzles to “assign” and “cross” information from one to the other, very “interleaved” information (and even considering the possibility of leaving some cages without a result or operator to make it more difficult). (This “replaced puzzles” have nothing to do with the actual twin puzzles where the numbers are in the same positions while the cages and operators vary).
