creating a more complex sudoku (69x6)

Question

I would like to know if its possible to create a "sodoku" with this rule: in a table $69\times 6$ i want to put in the numbers from $1$ to $46$ repeated $9$ times, each numbers HAS to stay in the same column with every other number $1$ and only $1$ time.

Meaning: if I have the $9$ columns that have the number $1$ in it, the other $5\times 9$ $(45)$ numbers will be all the numbers from $2$ to $46$ so that $1$ is with all numbers $1$ and only $1$ time.

(i made a program that tries to fill in the numbers but i was never able to put in more than $382$ of $414$ numbers) here it is: http://pastebin.com/JFtb8SVg

i didn't post it here because its $350$ lines, and commented in Italian, i don't so much want you to correct my program as much as tell me if what I'm trying to do is possible or not.

Thanks

Gokalex

Answer 1

You are looking for a $(46,6,1)$ block design. According to this paper they don't exist.

---

A few notes to help with reading the links. Wikipedia says:

a family of $k$-element subsets of $X$, called blocks, such that the number $r$ of blocks containing $x$ in $X$ is not dependent on which $x$ is chosen, and the number $\lambda$ of blocks containing given distinct points $x$ and $y$ in $X$ is also independent of the choices.

In your case, the set $X$ is the set of integers $\{ 1,2,\ldots, 46 \}$. The size of $X$ is called $v$, so $v=46$. The blocks are the columns, each of which has size $k=6$. The number of blocks containing an integer $x \in X$ is independent of $X$: namely, that number is $9$. Given distinct integers $x$ and $y$ in $X$, the number of blocks containing $x$ and $y$ is independent of the particular values of $x$ and $y$: namely, that number is $\lambda=1$.

Scrolling down, I read "The design is called a $(v, k, \lambda)$-design." So I googled for (46,6,1) block design.

Scanning through the early pages of the paper, I see the following on the second page:

The incidence matrix of such a design has 46 rows and 69 columns. In each row, there are 9 ones. In each column, there are 6 ones. Each pair of rows intersect in exactly one column.

So this is your construction. Instead of writing $6$ numbers in a columns, they write a column of height $46$ with $6$ ones in it.

The rest of the paper explained that this is the result of an 80,000 hour computation (presumably done in parallel, since that is 9 years), so I stopped reading.