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No. The problem has not been solved. Not at all. The lego model and the Mecano Model are not exact replicas.
From the Mecano replica site

A number of technical problems had to be overcome. First the representation of decimal digits - since standard Meccano is not well endowed with elements possessing a 10 fold symmetry, second finding a mechanism to reliably readout the value of a stored digit and add it to the adjacent column while retaining the original value, and finally organizing the overall drive mechanism such that the required torque could be distributed reliably with no slipping. The handling of carry propagation turned out to be one of the simpler aspects. Babbage was able to design specialized parts to fit his needs. In a sense, this makes the Lego and Mecano projects more admirable, but it's only with an accurate replica that historians can get a sense of

1. How fast would it have been?
2. How often would it have jammed?
3. Could it have been used for its intended purpose-- publishing tables that were more accurate than the competitors?

Those questions were answered with the first replica. The second model was bought essentially as a work of art for Mhyrvold's "living room."

Sometimes, in this digital age, we forget that precision engineering is necessary. Models and designs don't necessarily scale.

Consider a microscope, for instance. At the very lowest magnification, it's very basic-- a light source, possibly in the form of a mirror reflecting sunlight. A stage to hold the slide. A couple of lenses. A body to hold it all together. At higher magnifications, the light source needs to become more powerful and more reliable. The body of the microscope needs to be more rigid. Chromatic aberrations can obscure any additional detail, so you need special lenses. You might want a mechanical stage, so that you can zero in on the extra detail without jerking the slide about. You might want to put the microscope on an isolation platform if mechanical or building vibrations are disturbing the sample. A small improvement in signal to noise ratio may require extensive reengineering and considerably more expense.

Babbage was trying to design a system that was more precise and more reliable than what was otherwise available. It was not enough for him to design a toy system and assume it would scale up without jamming, without loss of precision, without breaking. He had to produce a working machine. In the end, he never really finished. The faithful replicas help us understand whether Babbage would have succeeded if he had continued to receive funding.

More images and links, courtesy of FreeCache. Due to Slashdot's lameness filter, I'm filling in some characters here so the character per line average goes up.

Enjoy a nice unsorted list of some images, courtesy of FreeCache. I wish more people would use this service in the future.

• RDA(Robinson's Differential Analyzer)
• Integrators
• Input Table
• Dual Output Table
• Interconnect Section

And some more links that the author is working on, apparently:

• Torque Amplifier
• Helical Gearbox

More images and links, courtesy of FreeCache. Due to Slashdot's lameness filter, I'm filling in some characters here so the character per line average goes up.

Enjoy a nice unsorted list of some images, courtesy of FreeCache. I wish more people would use this service in the future.

• RDA(Robinson's Differential Analyzer)
• Integrators
• Input Table
• Dual Output Table
• Interconnect Section

And some more links that the author is working on, apparently:

• Torque Amplifier
• Helical Gearbox