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I'd not seen that design before.  One issue I notice with it is the 220K series resistor into the ADC input.  There's a paragraph about why that was needed in your link above, entitled "220k output resistor" but I think it's misguided.   It really should be "10k or less" as stated in that pargraph.

But I doubt that's the cause of all of your problems.  The best bet is probably to debug the h/w first with a voltmeter.  Have the f/w select a single channel and repeatedly read and display that channel.  While that's running, make sure all the voltages are correct at the various stages and see if that voltage is consistent with what's being displayed.  Then repeat for each of the other channels.

I strongly suspect that either your circuit isn't what you think it is, or those blobs of solder have a whisker somewhere to an adjacent blob.

My first suggestion is to measure the voltages at all the multiplexer pins, and check that they are what you expect. If they are not, and of that I'm fairly certain, then:

Second is to use a continuity meter to check for shorts - everywhere that you have used pads adjacent to one another.

Third, reverse the 'bottom' photo, trace the 'tracks' and overlay that on the top photo, then check that against the circuit diagram. (Actually, it may be better to draw the diagram from what you actually have and compare that with the original.) I'm not at all convinced that you've got the correct pins for the multiplexer IC. Where do you think pin 1 is? I think it's nearest the op.amp, and that doesn't seem to fit your connections.

Fourth, next time you should invest in a roll of tinned copper wire and use that instead of all that solder, tacking it down at the ends and every 4th or 5th pad elsewhere where there isn't a component. I'd have used stripboard rather than what you have done.

I concur about the 220 kΩ. I think increasing noise when the value was reduced is a symptom of other problems - possibly power supply noise - that should be tackled. My analysis is that resistor plus strays were behaving as a low-pass RC filter.

Thank you for your prompt reply and advices. I'll try to apply all that to discover what is the problem. I tried to measure the voltage at the input of the multiplexer while the “probes” 1KOhm resistors are attached to input. Power source for the circuit is 5V, and every input of the multiplexer is showing me 0.45V, while the output from the multiplexer is showing me between 0.10V - 0.13V, like at the common output, so at each separate output. I have attached pictures for the mux and OpAmp. I am using ARTSCHIP CD4052BE M 0825 mux and TI 12CJ8YP LM324N OpAmp. I tried manually to select channels for the multiplexer, for example, I brought A channel to 5V, and B to GND, I tried all combinations. I noticed when I connect A and B to GND to get values for all 4 inputs between 200.00 and 500.00. For other combinations of A and B channels the value are 1024.00.

Links for mux and OpAmp that I'm using.

From the attached picture of the mux and OpAmp IC, you can see how I made the connections from the inputs to the mux inputs, and from the mux outputs to the OpAmp. From the mux pin 13 I went to OpAmp 3. I used OpAmp 2, and with OpAmp 3 went to the OpAmp 4, I did’t used the OpAmp 1 from the LN324N IC. I will try one more time to trace the "tracks" and go through the optical exam of the circuit.

Hy, thank You all for the answers. It's interesting that now my first "probe" is getting some values between 150.00 and 500.00, and other 3 "probes" are 1024.00. I tried with optical check of the traces and it is looking like there is no contact with neighbor track. I have measured the multiplexer input 0.45V, while the output from the multiplexer is showing me between 0.10V - 0.13V, like at the common output, so at each separate output. “probes” are 1KOhm resistors attached to input, power source for the circuit is 5V. The first problem, perhaps the only one begins from the mux output, because it does not follow input values.

Unfortunately I do not have a continuity tester. I have reversed the photos and analyzed the "tracks" and it seem to me that its ok, and I analyzed the connection (tracks) of my board with the shematic diagram. I think that the pin's for the mux are ok. I have attached pictures for the mux and OpAmp. I am using ARTSCHIP CD4052BE M 0825 mux and TI 12CJ8YP LM324N OpAmp. I tried manually to select channels for the multiplexer, for example, I brought A channel to 5V, and B to GND, I tried all combinations. I noticed when I connect A and B to GND to get values for all 4 inputs between 200.00 and 500.00. For other combinations of A and B channels the value are 1024.00.

Next time I will definitely use copper wire or the stripboard.

I will try to put all this together on a protoboard and see if I can get some values. I will post the pictures, and try to recycle the components from this board that I have soldered.

Of course any other advice is welcome.

Please read the page that the link "Read this before posting" points to. I've deleted all your posts except the ones above.

"Unfortunately I do not have a continuity tester."
If you want my honest opinion, you are wasting your time, and probably money too, without a reasonable multimeter. You will really struggle without one. There is a page in Resources|Building Blocks that should give you a guide. I would caution against buying a very cheap one, because you will soon find it is not good enough, one similar to the cheapest one in the table will be adequate for most things you need to do.