How does complex programming work on a computer chip that only has 300-500 connector pins on the underside?

There has been some incredibly complex programming done on processors with only 40 pins. A quarter of the pins on a modern processor are just power and ground pins anyway.

The pins are just mechanical inputs and outputs; there;s nothing innately smart about them.

The programming on the cip – the integrated circuit – is what makes the difference. With 500 mechnical connectors, a programmer can control lots of different things.

Morse Code only has one pin, and can communicate any verbal message you can imagine. Computers can do the same category of thing – it’s called using language. Having more than one pin, lets you communicate more information at the same time, but it does not really increase the complexity of what can be communicated.

Communication can be done in a number of ways. the methods include: Parallel – where several lines are all sending a part of the message at the same time (Imagine you want to send the number 1234. Parallel communication would have all 4 digits sent at the same time.) Serial communication has one line sending different messages at different times. (If you want to send the number 1234 it might send the 4 first the 3 next then 2 then 1.)
It is way more complicated than this. There are token rings, communication buses, phase modulated, watch dog timer synchronization,.. endless.
The architecture is carefully chosen to optimize the processor capability with the I/O needs of the system. Rest assured, every pin has been discussed and argued over by engineers and designers for many hours to come up with the layout we see.

Multiple chips?

My son has been working on these issues since grad school but damned if I understand it.

The “complex program” does not reside in the chip, The program instructions are fed, one at a time, to the chip from memory. The CPU chip processes one at a time, or just a few at a time in parallel pipelines.