 A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?
1. **Understanding the Problem:**
   We need to determine the minimum number of lottery tickets required to ensure that at least one ticket matches the winning permutation in at least one cell. The tickets are permutations of the numbers \(1, 2, 3, \ldots, 50\).

2. **Constructing a Winning Strategy:**
   We start by considering the construction of 26 tickets. Each ticket will have the last 24 numbers fixed as \(27, 28, \ldots, 50\). The first 26 numbers will be cyclic shifts of the sequence \([1, 2, \ldots, 26]\). This ensures that for any position \(i\) from 1 to 26, there is a ticket where the number \(i\) appears in the \(i\)-th position.

3. **Verification of the Strategy:**
   To verify that 26 tickets are sufficient, consider the following:
   - For each position \(i\) from 1 to 26, there is exactly one ticket where the number \(i\) appears in the \(i\)-th position.
   - For positions 27 to 50, the numbers \(27, 28, \ldots, 50\) are fixed and do not change across the tickets.
   - Therefore, any winning permutation will match at least one of these 26 tickets in at least one position.

4. **Proving 25 Tickets are Insufficient:**
   Suppose, for contradiction, that 25 tickets are sufficient to guarantee a win. For each position \(i\) from 1 to 50, let \(S_i\) be the set of numbers that never appear in the \(i\)-th position in any of the 25 tickets. Clearly, \(|S_i| \geq 25\) because each number can appear at most once in each position across the 25 tickets.

5. **Lemma and Proof:**
   - **Lemma:** For any subset \(S\) of \([50]\), \(|\cup_{e \in S} S_e| \geq |S|\).
   - **Proof:** If \(|S| \leq 25\), this is trivial because \(|S_e| \geq 25\) for all \(e \in [50]\). If \(|S| = t > 25\), since every number \(i\) can appear at most 25 times (once per ticket), there must exist some \(f \in S\) so that \(i \in S_f\). Otherwise, \(f\) would appear at least \(t\) times. This implies that \(|\cup_{e \in S} S_e| = 50\), proving the lemma.

6. **Application of Hall's Marriage Theorem:**
   Consider a bipartite graph where one set of vertices is \([50]\) and the other set is also \([50]\). There is an edge from vertex \(i\) to vertex \(j\) if \(j \in S_i\). By the lemma, this graph satisfies Hall's condition, implying there is a perfect matching. This perfect matching corresponds to a ticket that guarantees a loss, contradicting the assumption that 25 tickets are sufficient.

7. **Conclusion:**
   Since 25 tickets are insufficient and 26 tickets are sufficient, the minimum number of tickets required to guarantee a win is 26.

The final answer is \(\boxed{26}\)