Hi,

  I'm Rue Mohr, you might know me from such places as the internet, the information superhighway, or the world wide web.

Today I'd like to tell you about an adventure in optimization of a 7 segment decoder.

(7 segment display with the highly debated hook applied to the digit "7", OOOoooo)

For <lots of> decades people have fought with designing display decoders to take 4 bit BCD and generate 7 segment digits with it. Its not easy, there is a lot of decoding and re-coding involved.

(solution by the worlds greatest logic engineers, immortalized in the 7447 logic chip)

Granted, today we can just use an FPGA, CPLD, PAL, or 48MHz RISCV to perform this 4 bit to 7 segment decoding for us, but, is there a cheaper way?

(Could a way be found to take a few diodes and transistors and produce the coveted 7 segment patterns?)

My proposal was to ditch the BCD input. The particular application I have targeted for this (ok, yes, its a CLOCK...) does not NEED to be sending BCD to the displays. I do want the codes to be 4 bit to reduce wire count (otherwise, just sending the 7 segment patterns directly to the displays is easy enough)

(Super evil BCD bad boooooo, Hissss...)

Is there a 4 bit pattern that could more easily be converted to a 7 segment font than BCD? I decided to put a bounty* on the solution....

Now this puzzle has a few dimensions, there are 2 ways to make a 1, 6, 7, and 9 on a 7 segment display, that means there are 16 different 'fonts' that can be used. Also the choice of which 10 of the 16 4 bit codes to use is completely open.

(* the bounty was nothing, so there wasn't an overwhelming set of answers)

There were a number of answer that came back, one of them seemed unbelievably simple. (Don't get me wrong, I'd been working on a solution myself, but I'm REALLY BAD at this type of puzzle) 

The above equations resolve into only a little bit of circuitry. That became even less when @minusYCore pointed out an optimization I'd missed on the G segment.

Now, I can't prove this is the simplest decoder that can be made for 4 bit to 7 segment, but Its in the right order of magnitude. Part of what makes this tiny is the direct bit-to-segment relation of e and f.

(Only a few components to implement the logic worked out by @amohr)

At this point, the circuit had already been simulated by a few people, and, with quirks, confirmed to kinda work.

But I build them anyhow :]

(The decoder circuit is the top board)

Two small breadboards, the top one for the decoder, and the bottom one for the 4 bit input driver. Everything checked out. The next obvious step is to make a PCB that fits on the back of a 7 segment display, which, seems plausible...

(The "Train wreck" stage of designing a PCB)

Hours of kicad later.. (you don't really notice if go by) a PCB emerged. 

So, the PCB has been ordered, the results will come later. In the meantime its been a great adventure. It even has hack codes (the 6 unused codes cause the logic to do 'other' segment patterns)

I would also like to give mention to work done by (twitter) @poulsfriend, @oric_iss, @likimmo.

 This was another answer, it does resolve down to a bit more circuitry. (but its really close)

But... Why not use a Greenpack? CPLD? FPGA? or RISCV microcontroller?

THIS circuit, is cheaper, its 8 resistors (which you need anyway) and a total of less than 20c in discrete components. (at shipped, hobby volume prices)

It has not been proven to me yet, that you can actually GET (with shipping) a 10c microcontroller FOR 10c. (yes, I know there have been a lot of articles written about this) You tell me the source, and I'll tell you the price they give me (yes, unit price with shipping included)

 Rue M.