Step 1: Identify the three alkynes.
I: C2H2 (acetylene, HC$\equiv$CH) - terminal alkyne with no alkyl substituents. II: (CH3)2C2 (but-2-yne, CH3-C$\equiv$C-CH3) - internal alkyne, no terminal H. III: CH3C2H (propyne, CH3-C$\equiv$C-H) - terminal alkyne with one methyl group.
Step 2: Recall the acidic character of alkynes.
Terminal alkynes ($\equiv$C-H) are acidic because the sp carbon is highly electronegative (~50% s-character). The sp C-H bond has higher s-character than sp$^2$ (33%) or sp$^3$ (25%), making the bond stronger and the proton more easily released. Internal alkynes have no terminal C-H, so they have essentially zero acidic character.
Step 3: Compare I (C2H2) and III (CH3C2H).
Both are terminal alkynes. Acetylene (I) has no electron-donating groups. Propyne (III) has a methyl group attached to the sp carbon, which donates electrons (through induction), making the C-H bond slightly less polarized and thus LESS acidic than acetylene. So I is more acidic than III.
Step 4: Place II (internal alkyne, no terminal H).
But-2-yne (II) has NO terminal alkyne H at all. Its pKa is approximately 44 (essentially non-acidic), much less acidic than terminal alkynes (pKa ~25). So II is least acidic.
Step 5: Arrange in increasing order of acidity.
Least acidic to most acidic: II < III < I. (Internal alkyne < methyl-substituted terminal alkyne < unsubstituted acetylene).
Step 6: Final answer.
Increasing order of acidic strength: (CH3)2C2 < CH3C2H < C2H2, i.e., II < III < I.
\[ \boxed{II < III < I \text{ (option 2)}} \]