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Number of grids
https://www.calcudoku.org/forum/viewtopic.php?f=16&t=1121
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Author:  jpoos  [ Mon Jan 14, 2019 11:32 pm ]
Post subject:  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 [biggrin]

Author:  oldmathtchr  [ Tue Jan 15, 2019 8:03 am ]
Post subject:  Re: Number of grids

My first impression is 288: 4! x 3! x 2! x 1!

Author:  pnm  [ Tue Jan 15, 2019 9:57 am ]
Post subject:  Re: Number of grids

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

[blink]

Author:  paulv66  [ Tue Jan 15, 2019 10:08 am ]
Post subject:  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.

Author:  pnm  [ Tue Jan 15, 2019 10:12 am ]
Post subject:  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 [smile]

Author:  paulv66  [ Tue Jan 15, 2019 11:11 am ]
Post subject:  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 [smile]


Here you go - https://en.m.wikipedia.org/wiki/Latin_square

Author:  jaek  [ Tue Jan 15, 2019 2:15 pm ]
Post subject:  Re: Number of grids

pnm wrote:
I've been thinking on and off about the question: how many possible 4x4 Calcudoku puzzles are there?

[blink]

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.

Author:  pnm  [ Tue Jan 15, 2019 2:59 pm ]
Post subject:  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..

Author:  oldmathtchr  [ Tue Jan 15, 2019 3:03 pm ]
Post subject:  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

Author:  beaker  [ Tue Jan 15, 2019 9:51 pm ]
Post subject:  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??

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