Nair and Yuli play the following game:
$1.$ There is a coin to be moved along a horizontal array with $203$ cells.
$2.$ At the beginning, the coin is at the first cell, counting from left to right.
$3.$ Nair plays first.
$4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right.
$5.$ The winner is the one who reaches the last cell first.
What strategy does Nair need to use in order to always win the game?

1. **Initial Position and First Move**:
   - The coin starts at cell 1.
   - Nair plays first and can move the coin 1, 2, or 3 cells to the right.
   - To ensure a winning strategy, Nair should move the coin 3 cells to the right on the first move, placing it at cell 4.

2. **General Strategy**:
   - After Nair's first move, the coin is at cell 4.
   - The key to Nair's strategy is to always move the coin to a cell that is 3 cells ahead of a multiple of 4, i.e., cells of the form \(4k + 3\).

3. **Opponent's Move and Response**:
   - Suppose Yuji moves the coin \(n\) cells to the right, where \(n\) can be 1, 2, or 3.
   - Nair should then move the coin \(4 - n\) cells to the right.
   - This ensures that after Nair's move, the coin is always at a cell of the form \(4k + 3\).

4. **Mathematical Justification**:
   - If the coin is at cell \(4k + 3\) after Nair's move, and Yuji moves it \(n\) cells to the right, the new position of the coin is \(4k + 3 + n\).
   - Nair then moves the coin \(4 - n\) cells to the right, so the new position becomes:
     \[
     (4k + 3 + n) + (4 - n) = 4k + 7
     \]
   - Notice that \(4k + 7\) can be rewritten as \(4(k + 1) + 3\), which is of the form \(4m + 3\) where \(m = k + 1\).

5. **Winning the Game**:
   - The game ends when the coin reaches cell 203.
   - Since 203 is of the form \(4 \times 50 + 3\), Nair can always ensure that the coin reaches cell 203 on his turn by following the strategy of always moving the coin to a cell of the form \(4k + 3\).

Conclusion:
Nair's strategy is to always move the coin to a cell of the form \(4k + 3\). By doing so, he ensures that he will be the one to move the coin to cell 203 and win the game.

The final answer is Nair should always move the coin to a cell of the form \( \boxed{ 4k + 3 } \).