An application for zeroless squares

[Paul C. Leopardi]

Hi Matthjs,
I don't understand your puzzle. Maybe it is not completely specified below?
Thanks

On Fri, 29 Apr 2005 03:56 pm, Matthijs Coster wrote:
> Hello Seqfans,
>
> A month ago there was a discussion about zeroless squares.
> If you are interested read the (recreational) application below.
>
> Suppose you have to solve a crossword-puzzle of n x n.
> You have to fill it by the digits 0,...,9 and all the descriptions are
> equal, namely square of a number. How would you solve this
> crossword-puzzle, let's say crosssquare-puzzle?
>
> Important for solving this puzzle is to omit zeros at the first position.

Any number can be repesented using the digits 0 to 9. So are you asking for 
any square number in any position of your n x n square, or are there any 
other constraints?

> Well the 1 x 1 crosssquare-puzzle is easy. Only 1, 4, 9 are the solutions.

Why are 16, 25 etc. not solutions? Do you mean you only want single digit 
squares? Then only 1, 4 or 9 can be in any position of the n x n square.

> If you are interested:
> number of solutions of the 2x2 puzzle is 4,

Are any of the five following 2 x 2 squares:

1 1
1 1

1 1
1 4

1 1
1 9

4 4
4 4

4 4
4 9

solutions?

Do you mean that you have no repetition? Then you would need to allow more 
than 1, 4 and 9 in each position.

Are any of the six following 2 x 2 squares:

1  4
9 16

1  9
4 16

 1 4
16 9

1 16
4  9

 1 9
16 4

1 16
9  4

solutions?

Do you mean that each position in the n x n square is a digit of an n digit 
number read across or downwards, and that each n digit number must be a 
square?

Then are any of the six following  2 x 2 squares:

0 1
1 6

0 4
4 9

1 6
6 4

3 6
6 4

6 4
4 9

8 1
1 6

solutions? Are only the last four solutions because they do not contain 
numbers with leading zeros? Is this why you say there are only four solutions 
for 2 x 2?

Without further specification of the puzzle, I can't understand the rest of 
your message.

Also, what happens in number bases other than ten, for example in binary?

> 3x3 : 13
> 4x4 : 14
> 5x5 : 76
> 6x6 : 40
> 7x7 : 459
>
> So therefore I introduce the sequence of number of solutions:
> 3, 4, 13, 14, 76, 40, 459, ...
>
> What will happen when n grows? I don't know. Important is the last position
> where only 0,1,4,5,6,9 can be placed. Therefore you have to find squares
> with only these digits. How do the number of squares behave?