Step 1: Recall what the Lucas test actually measures.
The Lucas reagent is a mix of conc. $HCl$ and anhydrous $ZnCl_2$, and it works by turning an alcohol into an alkyl chloride through an $S_N1$ pathway. Since alkyl chlorides do not dissolve in this mixture, the solution turns cloudy the moment enough of them form.
Step 2: Think about carbocation stability, not the alcohol itself.
Because the mechanism goes through a carbocation, the speed of the whole test depends only on how stable that carbocation is: \[ 3^\circ \text{ carbocation} \gt 2^\circ \text{ carbocation} \gt 1^\circ \text{ carbocation} \] A tertiary carbocation forms almost instantly, a secondary one takes several minutes, and a primary one barely forms at room temperature at all.
Step 3: Sort the four given alcohols by degree.
2-methyl-1-propanol and butan-1-ol are both primary, butan-2-ol is secondary, while 2-methyl-2-propanol is tertiary since its $-OH$ carbon is attached to three other carbons.
Step 4: Match degree to observed speed.
Only the tertiary alcohol reacts fast enough to cloud the solution right away: \[ (CH_3)_3COH + HCl \xrightarrow{ZnCl_2} (CH_3)_3CCl\downarrow + H_2O \] \[ \boxed{\text{2-methyl-2-propanol}} \]