Using the variable $n$, the following expressions were written:

$$
2023 n, \quad n^{2}+n+23, \quad 3 n^{3}, \quad 10 n^{2}+2023
$$

An expression is called flattery if for every natural number $n$, the value of the expression is odd. Determine which of the four given expressions are flattery, and explain why. 

(L. Hozová)
In the following, we will suffice with known facts about the sums and products of even $(s)$ and odd $(l)$ numbers, which we briefly remind in the following tables:

| + | $s$ | $l$ |
| :---: | :---: | :---: |
| $s$ | $s$ | $l$ |
| $l$ | $l$ | $s$ |


| $\cdot$ | $s$ | $l$ |
| :---: | :---: | :---: |
| $s$ | $s$ | $s$ |
| $l$ | $s$ | $l$ |

- The first expression $2023 n$ is the product of an odd number $\mathrm{s} n$, so the result depends on the parity of $n$. The value of this expression is therefore not always odd (for example, for $n=2$ we get 4046).
- In the second expression, $n^{2}+n$ is always an even number: for $n$ even, respectively odd, it is the sum of two even, respectively two odd, numbers. The remaining addend is an odd number, so for any $n$ the value of the expression $n^{2}+n+23$ is always odd.
- The third expression $3 n^{3}$ is the product of an odd number $\mathrm{s} n^{3}$, so the result depends on the parity of $n$. The value of this expression is therefore not always odd (for example, for $n=2$ we get 24).
- In the fourth expression, $10 n^{2}$ is always an even number, since 10 is an even number. The remaining addend is an odd number, so for any $n$ the value of the expression $10 n^{2}+2023$ is always odd.

The odd-producing expressions are $n^{2}+n+23$ and $10 n^{2}+2023$.

Evaluation. 1 point for each correct and justified answer; 2 points for overall quality of the commentary.

Note. The evenness of the second part of the second expression also follows from the following transformation: $n^{2}+n=n(n+1)$. On the right side, it is the product of two consecutive numbers, thus the product of an even and an odd number.