How To Build a "Ping-Pong" Using any of (8251, 8255, LED'S)?

"No microcontrollers are allowed."

What's the point of this exercise? The 825x devices you mention in the thread title are designed to work with microcontrollers/microprocessors.

To design this system these days without using a microwhatever would be idiotic. Is this a college project?

As weird as it sounds, Microprocessor (8088) is allowed, microcontrollers are not! The main point is to use as simple and "old-fashion" techniques as possible.

OK, got it! I'll have a think.

Away to Italy tomorrow for a couple of days, so don't hold your breath!

I suggest using a shift register based circuit. Standard chips provide Left & Right shifts, simple logic can detect when the pulse is at the last LED and then lach the direction to move it backwards.

Alternatively, you could load the LED pattern for 10 bounces into a ROM 80, 40, 20, 10, 08, 04, 02, 01, 02 ,04 ,08 ,10 etc (hex numbers) and clock through the address. If there is no "return" signal when the output is 80 or 01h the jump to the end.

There are many other posibilities even without a micro. What about doing it in analog?

This sound like a fun school project.