Is it possible to solve for any resistar values or all four?

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@LorisElec^^ : It doesn't appear that you are wanting to actually get help, but rather just have someone spoon-feed you answers. Maybe that's the case and maybe it's not.

In any event, both for your education and others that might stumble upon this thread, let's consider the problem in general conceptual terms.

If someone designs the circuit above, clearly they can pick any arbitrary values they want for R1, R2, R3, and R4 -- their choices are fundamentally independent, meaning that knowledge of what they chose for one value tells us nothing about what they chose for any of the others since they have absolutely no constraints on them.

Had we been given those values -- which is what we are used to -- then we would have had all the information needed to determine all of the voltages and currents in the circuit. Since they didn't give us that, we need an additional equation, above and beyond what we would normally have had, for each additional unknown. So four more equations are needed.

If we are going to determine which four values they picked, we need to be given information that applies constraints on the options available. Since we have four independent unknowns, we need four independent constraints.

Looking at the diagram, we note that there are three voltages annotated on it. These provide three of those constraints -- IF they are independent. Determining if they are independent can be easier said than done. For instance, we are told that one of the voltages is 5.94 V. This could have been a design constraint given to the original circuit designer. Are the other two voltages independent of this constraint? If we just ask if we are still free to choose whatever voltages we want at the other two nodes as constraints for the designer, we see immediately that the 5.94 V places a constraint on what we can pick for either of those two voltages -- for example, I couldn't pick one of them to be 7.5 V. But having a constraint placed on further choices as a requirement for keeping the circuit consistent does not necessarily mean that the choices aren't mathematically independent.

In this case, look at the voltages in any order and you see that, while each choice places bounds on the voltages you can pick for the next one in order to be consistent with the conditions imposed by the circuit, you are free to pick any voltage within those limits. That's a strong indicator of independence.

So that gives us three constraints with which to build three of our four additional equations.

Now what about the additional information written below the diagram? Does that give us additional constraints?

Unfortunately not.

Each of those three pieces of information is completely determined by knowing the voltage at the first node is 5.94 V.

The total current is simply the voltage across the 20 Ω resistor divided by that resistance.

The effective resistance of the network of unknown resistors is simply the voltage across them divided by the total current.

The total resistance is simply the sum of the 20 Ω resistance and the effective resistance of the unknown resistor network.

So we are left one equation shy.

Now, if the problem statement is accurately reflected by the last line on the diagram, the problem is solvable. There is no need for the solution to be unique in order for you to find a solution and present it as the answer.

 

@LorisElec^^ : It doesn't appear that you are wanting to actually get help, but rather just have someone spoon-feed you answers. Maybe that's the case and maybe it's not.

In any event, both for your education and others that might stumble upon this thread, let's consider the problem in general conceptual terms.

If someone designs the circuit above, clearly they can pick any arbitrary values they want for R1, R2, R3, and R4 -- their choices are fundamentally independent, meaning that knowledge of what they chose for one value tells us nothing about what they chose for any of the others since they have absolutely no constraints on them.

Had we been given those values -- which is what we are used to -- then we would have had all the information needed to determine all of the voltages and currents in the circuit. Since they didn't give us that, we need an additional equation, above and beyond what we would normally have had, for each additional unknown. So four more equations are needed.

If we are going to determine which four values they picked, we need to be given information that applies constraints on the options available. Since we have four independent unknowns, we need four independent constraints.

Looking at the diagram, we note that there are three voltages annotated on it. These provide three of those constraints -- IF they are independent. Determining if they are independent can be easier said than done. For instance, we are told that one of the voltages is 5.94 V. This could have been a design constraint given to the original circuit designer. Are the other two voltages independent of this constraint? If we just ask if we are still free to choose whatever voltages we want at the other two nodes as constraints for the designer, we see immediately that the 5.94 V places a constraint on what we can pick for either of those two voltages -- for example, I couldn't pick one of them to be 7.5 V. But having a constraint placed on further choices as a requirement for keeping the circuit consistent does not necessarily mean that the choices aren't mathematically independent.

In this case, look at the voltages in any order and you see that, while each choice places bounds on the voltages you can pick for the next one in order to be consistent with the conditions imposed by the circuit, you are free to pick any voltage within those limits. That's a strong indicator of independence.

So that gives us three constraints with which to build three of our four additional equations.

Now what about the additional information written below the diagram? Does that give us additional constraints?

Unfortunately not.

Each of those three pieces of information is completely determined by knowing the voltage at the first node is 5.94 V.

The total current is simply the voltage across the 20 Ω resistor divided by that resistance.

The effective resistance of the network of unknown resistors is simply the voltage across them divided by the total current.

The total resistance is simply the sum of the 20 Ω resistance and the effective resistance of the unknown resistor network.

So we are left one equation shy.

Now, if the problem statement is accurately reflected by the last line on the diagram, the problem is solvable. There is no need for the solution to be unique in order for you to find a solution and present it as the answer.

Hi,

I didn't know that the integer solutions were proved to be non existent that's interesting.
In the case of this circuit I think it is easier to prove because the equations for the resistors do not allow an integer solution due to the coefficients and the relationships between them. We can look at this in more detail later.

I think the simplest and most straightforward method now is to just do a symbolic nodal analysis. This provides variables for all the voltages for each node like v1, v2, v3. We can then set those voltages to the three voltages given in the problem statement, and immediately find the results for three of the four resistors, allowing one to be arbitrarily chosen or chosen based on some other criterion such as the minimum value, which turns out to be very interesting.
Anyway, once the three voltages are substituted for the three variables v1, v2, v3, the resulting simple equations for three of the unknown resistor values come out without even considering the required total parallel resistance of the four unknown resistors. The three voltages seem to be enough to specify the problem, and it sort of makes sense because we can only calculate three of the four resistor variables, and the impedance is sort of built into the 5.94 voltage level and the 20 Ohm resistor.
Nodal analysis strikes again

 

Notice that the voltages at the midpoints of the R0,R1 string and the R2,R3 string are both in the vicinity of 3 volts. 3 volts is about 1/2 of 5.94 volts; this means that the ratios of the R0,R1 combo and the R2,R3 combo are about 1/2 plus or minus a bit. The resistance of the R0,R1 series combo and the R2,R3 series combo in parallel is about 30 ohms, and if all 4 resistors were 30 ohms, the series/parallel combo resistance would be 30 ohms and the ratio of the two voltage dividers would be 1/2. It's possible then that the values of the 4 resistors are all near 30 ohms. Just noticing the preceeding bits is a hint.

It has been pointed out that there are actually an infinite number of solutions, but if the solution is limited to only integer values for the resistors, one might reasonably guess that there is only one solution; consider this a hint.

Hi,

I'd like to hear your thoughts on finding integer values even if we can't find any. Also to consider are near-integer values off only by some small error limit like 0.001 or even 0.01 or something.

 

Hard enough that there is no general algorithm to determine if a polynomial equation has integer solutions even when the coefficients are integers. It took seventy years, but it's been proven that no such algorithm can exist.

This might be relevant if the only route to an integer solution of the TS's problem required obtaining an integer solution of a polynomial equation by means of some general algorithm.

 

Hi,

I'd like to hear your thoughts on finding integer values even if we can't find any. Also to consider are near-integer values off only by some small error limit like 0.001 or even 0.01 or something.

There's always the brute force/exhaustive search method. We live in an era of great computing power. Write a program loop to choose an integer value for R0 from 25 to 45. Calculate the corresponding values for R1,R2,R3 and perturb them to the nearest integer values. Plug those values into the circuit and solve for the 3 pertinent voltages. When the voltages you get are properly rounded to 2 digits after the decimal point, and then are the same as the voltages annotated on the circuit of post #1, you have an integer solution.

 

I didn't know that the integer solutions were proved to be non existent that's interesting.

That's not what I said. The original question, Hilbert's Problem #10, was to devise an algorithm to determine whether or not a polynomial equation with integer coefficients had integer solutions.

This problem was shown to be undecidable. That does not mean that such equations don't exist, and for simple examples, there are algorithms to find the solutions. But in the general case, there can't be.

[/QUOTE]

 

There's always the brute force/exhaustive search method. We live in an era of great computing power. Write a program loop to choose an integer value for R0 from 25 to 45. Calculate the corresponding values for R1,R2,R3 and perturb them to the nearest integer values. Plug those values into the circuit and solve for the 3 pertinent voltages. When the voltages you get are properly rounded to 2 digits after the decimal point, and then are the same as the voltages annotated on the circuit of post #1, you have an integer solution.

Hello,

Yes, but that's a rather vague description. I thought you had something concrete in mind because you stated that you could find an integer solution.

I was asking partly because myself and WBahn presently believe there is no integer solution for all four resistors, and partly because I thought maybe you knew a trick.

I base my opinion on the "no integer solution" on the resulting equations that come from an analysis such as with Nodal analysis. To get an integer solution, you'd have to satisfy equations that have non integer coefficients, and they are different for at least three of the resistors.
For an arbitrary example, say we find this solution for R1:
R1=(124*R0)/173
Now if we make R0 a whole multiple of 173, or R0=173*a where a is an integer, then we may have these two left to look at:
[-(937660*a)/(6920-70238*a),-(1117580*a)/(6920-70238*a)]
Both of those would also have to be integers using the same value for 'a' as for R1.
R3 would then become:
R3=(323*R2)/271
and so R2 would have to be a multiple of 271, but it still has to satisfy the first value above -(937660*a)/(6920-70238*a).
Is that possible?
We also still have to maintain that required total parallel resistance and we could easily exceed that value.

 

That's not what I said. The original question, Hilbert's Problem #10, was to devise an algorithm to determine whether or not a polynomial equation with integer coefficients had integer solutions.

This problem was shown to be undecidable. That does not mean that such equations don't exist, and for simple examples, there are algorithms to find the solutions. But in the general case, there can't be.

[/QUOTE]
Hi,

Yes I did not state my reply correctly.

 

Hello,

Yes, but that's a rather vague description. I thought you had something concrete in mind because you stated that you could find an integer solution.

I was asking partly because myself and WBahn presently believe there is no integer solution for all four resistors, and partly because I thought maybe you knew a trick.

I base my opinion on the "no integer solution" on the resulting equations that come from an analysis such as with Nodal analysis. To get an integer solution, you'd have to satisfy equations that have non integer coefficients, and they are different for at least three of the resistors.
For an arbitrary example, say we find this solution for R1:
R1=(124*R0)/173
Now if we make R0 a whole multiple of 173, or R0=173*a where a is an integer, then we may have these two left to look at:
[-(937660*a)/(6920-70238*a),-(1117580*a)/(6920-70238*a)]
Both of those would also have to be integers using the same value for 'a' as for R1.

Per the forum rules, the TS is supposed to show some effort. I've been waiting a few days to see if he has given up because for a couple of days he wasn't even viewing the thread. I see now that he was here yesterday evening, but hasn't shown any work. So I'll wait another day or so.

I don't know any method for directly finding an integer solution. It's necessary to generate some possible integer solutions and test them. The procedure I gave is not at all vague. If there is an integer solution with R0 between 25 and 45 it will be found. However, one would like to find some technique to zero in on a solution with less brute force trial and error.

 

If there is an integer solution with R0 between 25 and 45 it will be found.

There is not and I see no reason to think there should be one.

I haven't shown the solution formula or the tabulated values behind my plot. It's homework, after all. But setting R0 to integer values to create the plot showed that R1 can be within 0.01 of an integer every 60Ω or so. R2 and R3 come close to an integer a few times over the 0-130Ω range of R0 I looked at.

The closest to an integer solution that I see is at R0 = 35Ω where R1, R2 and R3 calculate to 25.09, 26.02 and 31.01 Ω respectively.

 

There is not and I see no reason to think there should be one.

I haven't shown the solution formula or the tabulated values behind my plot. It's homework, after all. But setting R0 to integer values to create the plot showed that R1 can be within 0.01 of an integer every 60Ω or so. R2 and R3 come close to an integer a few times over the 0-130Ω range of R0 I looked at.

The closest to an integer solution that I see is at R0 = 35Ω where R1, R2 and R3 calculate to 25.09, 26.02 and 31.01 Ω respectively.

Everybody should keep in mind that an integer solution applied to the circuit in question will not give exactly the numerical values given in post #1. The exact values derived from solving the circuit with those integer values will have more than 2 digits after the decimal point. The values given in post #1 on the circuit itself will have been rounded to 2 digits after the decimal point, and the current will have been rounded to 3 digits.

The reason for thinking there is an integer solution is that exercises like this are created by humans (although nowadays it might have been created by an AI), and a human creator will almost certainly give the resistors values that are whole numbers.

How likely would it be that he (or she) would give them values like 35.0000000000, 25.086705202312, 26.0200693040716 and 31.012850129945? Which is what would give exactly (well, very very very close to) the values given in post #1.

 

There's always the brute force/exhaustive search method. We live in an era of great computing power. Write a program loop to choose an integer value for R0 from 25 to 45. Calculate the corresponding values for R1,R2,R3 and perturb them to the nearest integer values. Plug those values into the circuit and solve for the 3 pertinent voltages. When the voltages you get are properly rounded to 2 digits after the decimal point, and then are the same as the voltages annotated on the circuit of post #1, you have an integer solution.

But what limits R0 to just the range of 25 Ω to 40 Ω? What if the only integer solution required R0 to be 49 Ω? Or 6476362 Ω?

Since our node voltages are only spec'ed to three sig figs, and since we have an infinite number of real solutions that would hit those three values exactly, there are going to be an infinite number of integer solutions that satisfy the requirement to only produce those three voltages to three sig figs. It's this that makes finding integer solutions pretty easy -- in fact, by random trial and error I found one within just a few minutes.

If we had to find an integer solution (or whether or not one existed) that produced those three voltages perfectly, then that becomes a very difficult task with a huge search space. We have three arbitrary values (the fourth is dictated by those three). Even if we constrained ourselves to resistors under 10 MΩ, that's a brute-force search space of size 10^21. But that's doing it true brute force. In this case we have constraints that bring that down to "just" 10^7, and this is a reasonable task on today's machines (but it might yield no results yet if we had gone to 100 MΩ it might have).

If we had to find ALL integer solutions that produced those three voltages to three sig figs, that becomes a much harder problem because we have to explore the region around each "nominal" solution to identify the range of values of one of the other resistors that produces answers, and then the range of the third resistor for each of those, and finally the range of the fourth resistor for each of those.

 

I was asking partly because myself and WBahn presently believe there is no integer solution for all four resistors, and partly because I thought maybe you knew a trick.

I never said that I believed that there are no integer solutions, only that finding them may be easier said than done. At the very least, y have to constrain the search space arbitrarily, otherwise it is infinite. So you need to put an upper limit on what resistor value are allowed to be. There may be integer solutions above that, but you are artificially declaring that they don't count.

If we are only looking for integer solutions that result in approximate answers (i.e., match just to the values displayed on the diagram), then it becomes much more likely that MANY integer solutions exist,

As we require tighter and tighter matches to the annotated voltages, number solution space shrinks. At some point, it vanishes. Maybe that happens by the time we get to three sig figs, and maybe it doesn't. I don't know.

As an aside, what does it even mean for a resistor to have an integer value? Is 3.42 kΩ an integer-valued resistor? What about 42 mΩ. Fundamentally, there is nothing "integer" about a physical quantity that is continuous. This is like saying that 1 inch is an integer length but that 2.54 cm isn't, even though they are the SAME length, by definition.

What we are really saying is not that the resistance values are integers, but rather that the coefficients of the inferred unit scaling factor is an integer. In turn, what we are really getting at here is an implied assertion that whomever came up with the problem most likely chose such coefficients and we are hedging our bets by hoping that we can read their mind.

 

There is not and I see no reason to think there should be one.

I haven't shown the solution formula or the tabulated values behind my plot. It's homework, after all. But setting R0 to integer values to create the plot showed that R1 can be within 0.01 of an integer every 60Ω or so. R2 and R3 come close to an integer a few times over the 0-130Ω range of R0 I looked at.

The closest to an integer solution that I see is at R0 = 35Ω where R1, R2 and R3 calculate to 25.09, 26.02 and 31.01 Ω respectively.

I didn't look for values of R that were close to integer values (in ohms), but rather sets of integer-valued (in ohms) resistors that produced the three voltages to within 0.005 V.

I used R1 as my control resistance. Doing that I found a set of resistances that produced 5.943396 V, 3.226415, and 2.476415 V, which all round to the displayed values.

To show how sensitive things are, you used R0 as the control and ended up with nominal values for the others that are closed to integer-valued.

If I take your 25.09 Ω and use 25 Ω as my value for R1, I get values for R0, R2, and R3 of 34.88 Ω, 26.11 Ω. and 31.12 Ω.

If I use values for R0 to R1 of 35 Ω, 25 Ω, 26 Ω, and 31 Ω, I get the three voltages being 5.973700 V, 3.229167 V, and 2.4873958 V. That last one just barely misses the mark as it rounds to 2.49 V.

 

Per the forum rules, the TS is supposed to show some effort. I've been waiting a few days to see if he has given up because for a couple of days he wasn't even viewing the thread. I see now that he was here yesterday evening, but hasn't shown any work. So I'll wait another day or so.

I don't know any method for directly finding an integer solution. It's necessary to generate some possible integer solutions and test them. The procedure I gave is not at all vague. If there is an integer solution with R0 between 25 and 45 it will be found. However, one would like to find some technique to zero in on a solution with less brute force trial and error.

Hi,

Well what I meant was a more direct solution. A hunt algorithm is not what I consider a direct solution, and the way you originally worded it, it sounded like you had a more direct method in mind. It's ok if you don't though, but that was the impression I got from your post.
Along the same lines is, just shoot random values at it over and over again until you 'might' find a solution. That's a very simple method, but also not what I was hoping for. I had long ago proposed a search algorithm also, and so again I thought your ideas may have been better than that. Not a huge deal but that's what the impression was when you mentioned something like "discreet mathematics" or something.
Not a big deal though.
I was thinking if we could find limiting values we could search every integer in the book from 0 to those limiting values, but there are no limiting values unless you allow multiples of infinity, which I do not think helps here. The only limiting value I found so far was just a hair over 17 Ohms, and that puts a limit on the 'other' resistor due to the total parallel resistance constraint, but then the other two resistors seem to have no limit other than a limit on the ratio of the two.

Now as to a direct solution, I think there is one and I was pointing that out before how it might work. If I get a chance, I will look at that again. The whole problem is that if we multiply a whole number by a fraction like 1000/100 but then later are forced to use a fraction like 1000/99, we will never get an true integer solution.
I also think it is very possible we can't even get close to an integer solution for all four resistors, but I'll wait on that.
I also take the problem specification verbatim as to the voltages for example.

 

I never said that I believed that there are no integer solutions, only that finding them may be easier said than done. At the very least, y have to constrain the search space arbitrarily, otherwise it is infinite. So you need to put an upper limit on what resistor value are allowed to be. There may be integer solutions above that, but you are artificially declaring that they don't count.

If we are only looking for integer solutions that result in approximate answers (i.e., match just to the values displayed on the diagram), then it becomes much more likely that MANY integer solutions exist,

As we require tighter and tighter matches to the annotated voltages, number solution space shrinks. At some point, it vanishes. Maybe that happens by the time we get to three sig figs, and maybe it doesn't. I don't know.

As an aside, what does it even mean for a resistor to have an integer value? Is 3.42 kΩ an integer-valued resistor? What about 42 mΩ. Fundamentally, there is nothing "integer" about a physical quantity that is continuous. This is like saying that 1 inch is an integer length but that 2.54 cm isn't, even though they are the SAME length, by definition.

What we are really saying is not that the resistance values are integers, but rather that the coefficients of the inferred unit scaling factor is an integer. In turn, what we are really getting at here is an implied assertion that whomever came up with the problem most likely chose such coefficients and we are hedging our bets by hoping that we can read their mind.

Hello,

Ok you never said there were definitely no integer solutions, that's fine.

However, it's not much of a stretch to think that an integer solution would require whole number values in Ohms, which is how many other integer type solutions like this are stated. That means values like 1000, 1200, 1201, 5433, etc., just no fractional values like 43.2 or really even 43.001 unless you really want to specify an error limit like 0.001 or so.
It's not a stretch to believe that would be the typical integer interpretation.
However, following your suggestion, then you can decide yourself, but once you make that decision, just stick with it. For example, if you decide that 3k is integer but 3.3k is not integer, then just stick with that throughout. I would recommend sticking with the value in Ohms though rather than KOhms. You don't really have to post the actual solution either, just mention that you found one. We can go into detail at a later date.

I do not think there is such an integer solution exactly (zero error limit), which is what I was after. None of us has proven that there is none yet though (ha ha).

Thanks for your ideas too.

 

I also take the problem specification verbatim as to the voltages for example.

If you mean that you are considering the values in post #1 to be the exact solution values, I direct your attention to what WBahn said here:
https://forum.allaboutcircuits.com/...from-the-circuit-provided.196859/post-1861361

He said " Unless there is a problem-specific reason to do otherwise, the usual convention in engineering is to report answers to three significant figures. Intermediate results that are used in subsequent calculations should be reported with one or two additional sig figs so that the resulting round-off errors have little to no effect on the final result. It is also common to not count a leading 1 when counting sig figs. "

It's not conventional to think that when floating point values such as "3.23" are reported as the answer to a problem that they are exact. If one wants to provide an exact answer it should be a rational number like "155/48".

 

Hello,

Ok you never said there were definitely no integer solutions, that's fine.

However, it's not much of a stretch to think that an integer solution would require whole number values in Ohms, which is how many other integer type solutions like this are stated. That means values like 1000, 1200, 1201, 5433, etc., just no fractional values like 43.2 or really even 43.001 unless you really want to specify an error limit like 0.001 or so.
It's not a stretch to believe that would be the typical integer interpretation.
However, following your suggestion, then you can decide yourself, but once you make that decision, just stick with it. For example, if you decide that 3k is integer but 3.3k is not integer, then just stick with that throughout. I would recommend sticking with the value in Ohms though rather than KOhms. You don't really have to post the actual solution either, just mention that you found one. We can go into detail at a later date.

I do not think there is such an integer solution exactly (zero error limit), which is what I was after. None of us has proven that there is none yet though (ha ha).

Thanks for your ideas too.

While not definitive, I've explored the values for R1 from 13 Ω (it's trivial to show that R1 must be no less than 12.2 Ω) up to 1 MΩ in 1 Ω steps. For each value I calculated the nominal values (to something like 14 sig figs) for the other three resistors. I then rounded those to the nearest integer value. I then determined the maximum error in the three voltages and kept only those that were no more than 5 mV. I only found one. If I only required the results to match within 10 mV, after rounding to three sig figs, there were 23 solutions.

But I can't say that this approach truly looked at all viable solutions. It rounded each resistor value independently to the nearest ohm, but it's possible that the error introduced by rounding one or two of them in one direction might have been better offset by rounding another in the "wrong" direction. It wouldn't be to hard to modify my script to fully explore the small region around each nominal solution -- I think looking at all combinations of the floor and ceiling values (so eight combinations) would likely do it. While I think it's possible that this might reveal additional solutions, I would bet against it, if I had to place a bet.

As for the question of whether stopping at 1 MΩ potentially rules out any solutions, I think a careful analysis could be done to establish an upper bound for the resistor values such that changes could no longer move the parallel combination of the network enough to have any impact of significance. I'm pretty sure that 1 MΩ is well beyond that limit, but I'm not positive. Based on a quick survey of the behavior, I think the upper bound is probably closer to 2 kΩ.

 

I base my opinion on the "no integer solution" on the resulting equations that come from an analysis such as with Nodal analysis. To get an integer solution, you'd have to satisfy equations that have non integer coefficients, and they are different for at least three of the resistors.

But why do you say that they have non-integer coefficients? Is it because some of the information (the node voltages) don't look like integers? What if they had been written as 5940 mV instead of 5.94 V? Would that have made them integers?

Looking at the voltage divider constraint on R0 and R1, we have the equation

248·R0 - 346·R1 = 0

I'm pretty sure all of the relevant equations can be written with integer coefficients.

 

If you mean that you are considering the values in post #1 to be the exact solution values, I direct your attention to what WBahn said here:
https://forum.allaboutcircuits.com/...from-the-circuit-provided.196859/post-1861361

He said " Unless there is a problem-specific reason to do otherwise, the usual convention in engineering is to report answers to three significant figures. Intermediate results that are used in subsequent calculations should be reported with one or two additional sig figs so that the resulting round-off errors have little to no effect on the final result. It is also common to not count a leading 1 when counting sig figs. "

It's not conventional to think that when floating point values such as "3.23" are reported as the answer to a problem that they are exact. If one wants to provide an exact answer it should be a rational number like "155/48".

Hi,

Yes that is fine too, so I guess at this point it is time to divide the problem down into two or more modes.
The first is to find pure integers, that is whole numbers in Ohms for all four resistors, and the second
is to find approximations. Those would be the two main categories, and you could divide them further if you feel the need.

For me I started to look for the pure integer solution with whole numbers in Ohms just to see if there are any. I started to do this because if we can find this solution then every other subdivision should be somewhat moot. But if not that, then a proof that there is no such solution, which I think is possible. So for now, that's my goal.