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jpoos
Posted on: Mon Jan 14, 2019 11:32 pm
Posts: 146 Joined: Sun Nov 03, 2013 10:28 pm

Number of grids
Just recently, someone offered me the following question: How many ways are there to fill in a 4x4 calcudoku grid? That is: How many ways are there to fill in a 4x4 grid with the numbers 1 to 4, such that every row and column contain each number once (permutations are allowed)? What about 5x5 grids? Or n by n grids? I'm quite confident I've found the answer for 4x4 grids, but it isn't very elegant and doesn't easily generalise to bigger grids. This leaves me to wonder if there are nice ways answer these questions. I haven't given it too much thought, but I certainly will. So if anyone wants a puzzle, I challenge you to take a stab at this one




oldmathtchr
Posted on: Tue Jan 15, 2019 8:03 am
Posts: 16 Joined: Sun Mar 11, 2012 8:48 pm

Re: Number of grids
My first impression is 288: 4! x 3! x 2! x 1!




pnm
Posted on: Tue Jan 15, 2019 9:57 am
Posts: 2636 Joined: Thu May 12, 2011 11:58 pm

Re: Number of grids
I've been thinking on and off about the question: how many possible 4x4 Calcudoku puzzles are there?




paulv66
Posted on: Tue Jan 15, 2019 10:08 am
Posts: 609 Location: Dublin, Ireland Joined: Tue Mar 01, 2016 10:03 pm

Re: Number of grids
While it's an interesting question, there's not a straightforward solution. If you look up Latin square on Wikipedia, you will find considerable detail on the mathematics. I'd post a link, but I don't know if that's permitted.




pnm
Posted on: Tue Jan 15, 2019 10:12 am
Posts: 2636 Joined: Thu May 12, 2011 11:58 pm

Re: Number of grids
paulv66 wrote: While it's an interesting question, there's not a straightforward solution. If you look up Latin square on Wikipedia, you will find considerable detail on the mathematics. I'd post a link, but I don't know if that's permitted. Fine to post a link of course




paulv66
Posted on: Tue Jan 15, 2019 11:11 am
Posts: 609 Location: Dublin, Ireland Joined: Tue Mar 01, 2016 10:03 pm

Re: Number of grids
pnm wrote: paulv66 wrote: While it's an interesting question, there's not a straightforward solution. If you look up Latin square on Wikipedia, you will find considerable detail on the mathematics. I'd post a link, but I don't know if that's permitted. Fine to post a link of course Here you go  https://en.m.wikipedia.org/wiki/Latin_square




jaek
Posted on: Tue Jan 15, 2019 2:15 pm
Posts: 261 Joined: Fri Jun 17, 2011 8:15 pm

Re: Number of grids
pnm wrote: I've been thinking on and off about the question: how many possible 4x4 Calcudoku puzzles are there? That is an interesting question. On top of the possible Latin squares are the possible cages overlaying the cells. Theoretically you can have 1 to 16 cages. And then each cage has an operation. 16 cages would mean each cage is a single cell and the operator is '=' for each cage. A single cage including all 16 cells is possible, except two problems arise. First,  and / wouldn't work as operations. Second, neither +40 nor x331776 provide sufficient information for a unique solution. And then there are a _lot_ of permutations in between.




pnm
Posted on: Tue Jan 15, 2019 2:59 pm
Posts: 2636 Joined: Thu May 12, 2011 11:58 pm

Re: Number of grids
First question would be how many possible puzzles there are (and you'd like to correct for symmetries too I think)
Next is how many of these have a single solution..




oldmathtchr
Posted on: Tue Jan 15, 2019 3:03 pm
Posts: 16 Joined: Sun Mar 11, 2012 8:48 pm

Re: Number of grids
Here is another link that shows how the answer to the number of 4x4 squares is determined http://www.maths.qmul.ac.uk/~pjc/comb/ch6s.pdf




beaker
Posted on: Tue Jan 15, 2019 9:51 pm
Posts: 703 Location: Ladysmith, BC, Canada Joined: Fri May 13, 2011 1:37 am

Re: Number of grids
So, if you are talking about a 4x4 grid, does the total number of possibilities change when you "add" in the 4 different procedures??........not a Mathematician, but Iwouldn't think so??




