Circuit Challenge 2

Consider the circuit shown below.
Circuit Challenge 3 -- R-2R Ladder

The Situation

In your youth you went into a surplus store and purchased a box of about 200 carbon resistors.  You got a good price, and then took them home.  At home you noticed that about half the resistors are identically marked (same color code).  The other half are identically marked with a different color code.  You were not able to read the codes, but your neighbor had an ohm-meter.  The result was that the resistors had the following two values:  R and 2R.

Some time later (and after purchasing your own ohm-meter) you began constructing the circuit shown above.  You soldered the resistors together, working from right-to-left.  You ended up by placing the last 2R element shown at the left end of the circuit above.  You had been measuring the resistance after the placement of each resistor, but you saw a pattern emerging, so you soldered rapidly until you had used up all your resistors.  You measured the final input resistance and realized you could have achieved the same result with fewer resistors.

Assuming there were 100 2R resistors and 101 R resistors, what is RIN?
You may compare your RIN with the actual RIN by clicking here.