Possible resistances from four resistors

High school · ●●●○○ 50 pts

Rebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistances

a

and

b

ohms, and combines them into one by one of the following methods: (i) connect them in series, producing a resistor of

a+b

ohms; (ii) connect them in parallel, producing a resistor of

\frac{ab}{a+b}

ohms; (iii) short-circuit one of the two resistors, producing a resistor of either

a

b

ohms. Suppose that after three minutes, Rebecca has a single resistor with resistance

R

ohms. How many possible values are there for

R