Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties:
$*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering.
$*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$.
$*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$.

(Proposed by Gerhard Woeginger, Austria)
1. **Understanding the Problem:**
   We need to determine the largest integer \( N \) such that there exists a \( 6 \times N \) table \( T \) with the following properties:
   - Every column contains the numbers \( 1, 2, \ldots, 6 \) in some ordering.
   - For any two columns \( i \neq j \), there exists a row \( r \) such that \( T(r, i) = T(r, j) \).
   - For any two columns \( i \neq j \), there exists a row \( s \) such that \( T(s, i) \neq T(s, j) \).

2. **Interpreting the Conditions:**
   - The first condition implies that each column is a permutation of \( \{1, 2, 3, 4, 5, 6\} \).
   - The second condition implies that any two columns must have at least one common element in the same row.
   - The third condition implies that any two columns must have at least one differing element in the same row.

3. **Constructing the Table:**
   - We need to find a set of permutations of \( \{1, 2, 3, 4, 5, 6\} \) such that any two permutations intersect (agree on some value) and differ on some value.

4. **Counting Permutations:**
   - There are \( 6! = 720 \) permutations of \( \{1, 2, 3, 4, 5, 6\} \).
   - We need to find the largest subset of these permutations such that any two permutations in the subset intersect.

5. **Maximizing the Number of Columns:**
   - Consider the set of permutations where the first element is fixed, say \( \sigma(1) = 1 \). There are \( 5! = 120 \) such permutations.
   - For any two permutations in this set, they will always intersect at the first element (which is 1).

6. **Verifying the Maximum:**
   - To show that 120 is maximal, consider the structure of permutations. If we split all permutations of \( \{1, 2, 3, 4, 5, 6\} \) into sets of six cyclic shifts, we can choose at most one permutation from each set to ensure that any two permutations intersect.
   - Since there are \( 720 / 6 = 120 \) such sets, the maximum number of permutations that can be chosen is 120.

Conclusion:
\[
\boxed{120}
\]