Consider an $n\times n$ sudoku puzzle $P$ that I give you as input, and you have to decide whether the puzzle $P$ is solvable. Formally, we consider the following language $\text{SUDOKU} = \{ \langle P\rangle: \text{$\langle P\rangle$ is a description of a solvable sudoku problem} \}$

Note that the language $\text{SUDOKU}$ contains puzzles $P$ of a non-constant size.

Now if I give you a puzzle $P\in \text{SUDOKU}$ and a solution $S$ for it: so I give you a pair $\langle P, S\rangle$, then you can verify in time polynomial in $|\langle P\rangle|$ whether $S$ is a correct solution for the puzzle $P$. The latter proves that the language $\text{SUDOKU}$ is in NP.

However, solving $\text{SUDOKU}$ means that given any puzzle $P$, you have to decide whether it has a correct solution $S$, which is sounds harder since you have to look for a correct solution $S$ (from an exponential number of candidate solutions).

In other words, on an intuitive level, checking a solution should be easier than suggesting a solution (unless P = NP).

Consider an $n\times n$ sudoku puzzle $P$ that I give you as input, and you have to decide whether the puzzle $P$ is solvable. Formally, we consider the following language $\text{SUDOKU} = \{ \langle P\rangle: \text{$\langle P\rangle$ is a description of a solvable sudoku problem} \}$

Note that the language $\text{SUDOKU}$ contains puzzles $P$ of non-constant size.

Now if I give you a puzzle $P\in \text{SUDOKU}$ and a solution $S$ for it: so I give you a pair $\langle P, S\rangle$, then you can verify in time polynomial in $|\langle P\rangle|$ whether $S$ is a correct solution for the puzzle $P$. The latter shows that we have a polynomial time verifier for $\text{SUDOKU}$ and thus proves that the language $\text{SUDOKU}$ is in NP.

However, solving $\text{SUDOKU}$ means that given any puzzle $P$, you have to decide whether it has a correct solution $S$, which is sounds harder since you have to "prove" whether a correct solution $S$ exists (from an exponential number of candidate solutions).

In other words, on an intuitive level, checking a solution should be easier than suggesting a correct solution (unless P = NP).