Math Is Fun Forum

random_fruit

Member

Registered: 2008-12-25

Posts: 39

Factorials squared (re KenKen game)

You have probably come across the KenKen game by now.

Tedious explanation: a square grid left empty except for part rows, part columns, sub-squares, etc, so that every square of the grid is 'covered' by one part-row, part-column, etc.  Each part-row, part-column, sub-square, etc., has in its top-left-most square a result and a simple arithmetic operator, one of add, subtract, multiply, divide.  The subtract and divide part-rows and part-columns can only be two cells big.  Also, each complete row and complete column must contain all the digits starting at 1 and upto the size of the grid, but in any order.  More complicated shapes, such as letter L are allowed.  The puzzle is to deduce the values in each blank cell so that all the arithmetic is satisfied and each row and column contains 1 to grid-size.

Here is one which you can only understand in a mono-space font: (cut-n-paste it)

+---+---+---+---+
|6* |12*        |
+   +---+---+---+
|   |6+ |3+     |
+   +   +---+---+
|   |   |3  |5+ |
+---+---+---+   +
|9+         |   |
+---+---+---+---+

(Mono-spaced that for you.)
which you can try to solve for yourselves.  (Answer below...)

What I was wondering, having solved several of these now, is how many possible combinations of layouts of digits are possible, ignoring the cells and arithmetic.  Placing numbers 1-n in a row obviously gives factorial n, written n!  Placing numbers in a grid (so that each row and column contains 1 to n exactly once) has more combinations.

A one-sided grid has one combination.  A two sided grid has two combinations.  (They are:
12
21

and
21
12)

I think the number of possible combinations, N, is given by:

N = (n!) * (n!)

for n=3 and n=4.  Can anyone develop (or quote) a generalised proof, where n>=2?  Or am I simply wrong on this?

And the answer to the little grid above is
2143
3412
1234
4321
and I hope that I haven't made a silly in drawing up and checking the puzzle works!

Last edited by mathsyperson (2010-03-13 11:55:29)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Factorials squared (re KenKen game)

Hi random_fruit;

Placing numbers in a grid (so that each row and column contains 1 to n exactly once) has more combinations.

I think you mean an n x n Latin square:

http://en.wikipedia.org/wiki/Latin_square

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

random_fruit

Member

Registered: 2008-12-25

Posts: 39

Re: Factorials squared (re KenKen game)

Thanks, bobbym - I've never heard of them being a class of "Latin Squares" before - but I see what you mean.  Thanks.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Factorials squared (re KenKen game)

Hi;

Glad to help. I didn't know there was no known formula for the number of a n x n Latin squares. At least that's what the page says.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.