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The interesting thing here isn’t “spreadsheet, but backwards” so much as “spreadsheet as a constraint system”. Classic spreadsheets are basically DAGs: data flows one way and a lot of UX assumptions (and people’s intuition) rely on that. As soon as you allow arbitrary cells to be solved for, you’re in “which variables are free?” land, and most of the confusion in this thread is really about degrees of freedom, not about the math.

One way to make this less surprising might be to flip the default: treat all cells as fixed unless explicitly marked as solver variables, and give a lightweight visualization of “these are the cells that will move if you edit this one.” That keeps the power of a general constraint solver while preserving the mental model spreadsheet users already have, and it opens the door to more serious use cases (financial models, physics, scheduling) without feeling like spooky action at a distance.

That's great feedback, thanks! I agree with you, but I don't want to flip the default because this is an experiment I made for fun, and the whole point is to lean in to the chaos a little bit. In a serious product the UX would definitely need a lot more work though.

Graphically, I really like the way autodesk makes sketches in fusion 360 blue until they are fully constrained, and then they are black. My intuition here is that you could color code “degrees of freedom” and “locked” states so that it was more intuitive.

The first example on the main page has a formula with two variables being updated from changing one value. The immediate question I have is if I change the output, where does the extra degree of freedom come from on the inputs? Does one stay locked in place? Unclear.

I am a huge fan of the concept though. It's been bugging me for years that my spreadsheet doesn't allow editing text fields after filtering and sorting them down to the subset I want. I have to go all the way back to the mess of unsorted input rows to actually edit them.

  Does one stay locked in place? Unclear.

It would make more sense to preserve the ratio if possible.

Yeah, this concept is interesting but the fact that the simplest test case gives what's fundamentally a surprising result is very annoying.

It also doesn't help that in the example, the expected outcome of 53.3333/46.6667 isn't even considered.

You can do this with bidicalc already! You just have to model the problem correctly. If you expect the ratio to remain constant, what you actually want is a problem with a single free variable: the scale.

    A1 = 1.0       // the scale, your variable
    A2 = 6 * A1    // intermediate values
    A3 = 8 * A1 
    A4 = A2 + A3   // the sum

To get that, you could pass the ratio explicitly. C = 5A + 7B

What is inputs a,b,c,d,and e are polynomial coefficients? I am hoping to get a fields medal plz respond.

You can actually solve fifth order polynomials with bidicalc! But it's a numerical solution, not an algebraic one, so no Fields medal.

I think it would be good if you could lock one of them.

You can.

With # (octothorpe) #Val

100% this. When I reached the end of that page I felt pranked because the obvious question was never answered. How are these cases resolved? Is it possible to fix some inputs and only update others? What if I sometimes want to change input A, and other times I want to update input B? All this should be explained as early as possible.

You can do it and it is explained, actually. Use # as a prefix to indicate a constant, e.g.: #50 will be a constant and not a variable.

In the future I'd like to support more user input constraints, in particular domain constraints for variables. So you could tell the solver that this cell must remain in some interval, and it would respect that interval instead of assigning any real value.

IMO constant should be the default and variables should be annotated.

I have wanted one general application of this idea in a spreadsheet. Specifically, I track some of my running, including speed (pace), distance, and time. Under different circumstances, I have exactly two of the three available and I want the third to be computed, but it varies which. I have found it fairly difficult to implement this kind of data entry in Google Spreadsheets and Excel, even know conceptually it's a very simple constraint "a*b=c" where I know some two variables.

As a more substantive comment: You may find the thesis "Propagation networks : a flexible and expressive substrate for computation" by Alexey Radul interesting. https://dspace.mit.edu/handle/1721.1/54635

You could create a table with 3 columns: distance, time, pace. Set the display format for time and pace to "Duration".

Enter these formulas:

  distance = time / pace
  time = distance * pace
  pace = time / distance

You just need two spreadsheet tabs. One for the "raw" input and one with a formula that either takes the input if it exists or falls back to the calculated version

Came here to see if anyone mentioned propagators. That thesis is excellent. I second the recommendation.

Can you enter an RSA key and have it produce two prime numbers?

A random tool like this would be the most entertaining possible way for something like that to be unleashed on the world

My brother once suggested that there are probably bits of code/algorithms that would be world changing if they were released in academic journals, but instead were written by some unknowing programmer in an afternoon for their job coding embedded systems for refrigerators.

This particular example may be unlikely, but it's a very fun idea.

An anonymous 4chan user once solved a 25 year old maths problem, to answer a question about the watch order of an anime. https://www.scientificamerican.com/article/the-surprisingly-...

Iirc, Heisenberg reinvented Matrix calculations to solve a problem in quantum physics. Not being a mathematician, he wasn't aware of the concept. Born recognized what Heisenberg had done and introduced him to his own reinvention.

PBS Space Time released a video with that story yesterday: https://www.youtube.com/watch?v=c-Q5r3THR3M

Lots of people working in different fields end up reinventing things that have been known to math for centuries, often in clunky roundabout ways. I imagine some of them figure out things not known to math, but it's far more likely to go the other way.

Folks shouldn’t be afraid to “rediscover” stuff.

Primarily because the learnings you make are the same as the original “discoverer”. Without those learnings, you might not be able to arrive at your true destination.

>Folks shouldn’t be afraid to “rediscover” stuff.

Luckily no one is suggesting that.

A lot of people suggest that. So many that it has become an idiom. "Don't reinvent the wheel."

> Lots of people working in different fields end up reinventing things that have been known to math for centuries

I remember reading, about a year or two ago, about a medical doctor that published a paper rediscovering calculus (I just looked it up, it happened in 1994, there’s been many articles and videos about it)

It's not clear from the Wikipedia article linked below whether she was rediscovering part of calculus or knowingly rebranding it. Do you know more details?

lmao: https://en.wikipedia.org/wiki/Tai%27s_model

This is such a great story

it's a fact of geographical and social independence.. so far there's no way to know what everybody did or is doing (well there's twitter but it's configured on noise rather than signal)

A lot of the time engineers are focussed with solving a problem, to build a working machine/program, while academics just want to publish.

This is also true with patents.

Jokes aside, let's say someone does figure out how to break RSA over a weekend project. The evil options are easy to come up with, but what is the actually responsible, ethical, thing to do? Never tell anyone?

Contact a known and trusted security researcher who can verify to the world that you did what you said you did, so everyone else can have as much time as possible to figure out exactly how fucked they are. Doing nothing isn’t an option; once someone figures something like that out, it signifies that conditions were ripe for the discovery to be made, and it’s only a matter of time before it’s discovered again independently.

Also fairly reasonable to assume it has already been done by someone who had a motive to break it and is keeping quiet.

Pretend you had developed a quantum computing advancement and push people to post quantum encryption

Migrating to post quantum encryption is important, but it's also important to not be herded into a "solution" that can be/has been easily compromised.

https://blog.cr.yp.to/20251004-weakened.html

In Prolog you can write rules (similar to functions in other languages) so that they work "both ways". Let's say you have this rule that defines how pace ("runner's speed") relates to distance and time:

  :- use_module(library(clpr)).

  pace(Km, Minutes, Pace) :-
    { Minutes = Km * Pace }.

You can query it, giving unknowns an uppercase `Name`, and it will give you the possible values for it:

  pace(5, 24.5, Pace)
  pace(40, Min, 5)
  pace(Km, 24.5, 5)
  pace(Km, Time, 5)
  

So if you had a rule that defines RSA key calculation this way, you could enter a key and get all valid solutions for the primes. But of course complex calculations still take a long time. I assume it's similar to a brute force attack in that way (Prolog has clever strategies to explore the solution space though).

Disclaimer: I'm not an expert in Prolog or cryptography, so this might not be 100% accurate.

Or enter a public key + some encrypted data to get the private key

In Excel you have goal seek for this functionality. I believe it does some form of numerical solving of the equation system.

Good for every situation when you need to solve equations!

In the context of using spreadsheets I think about solving simple financial or maybe construction/mechanical design problems where you don’t want to solve it manually or program it and a spreadsheet is a quick and useful interface.

This is very different in practice, because it is pervasive rather than something you have to set up for particular cases.

If this was usual it would help a lot with people's tendency to hard code the correct answer rather than fix formulae. Just that aspect of it would be a huge improvement. People do this all the time with simple financial problem, for example.

A lot of what people use spreadsheets for is not all that simple. Again, especially with financial applications. People manage huge amounts of money using horribly complex models implemented in Excel.

It does not. It perturbates the variables and uses a hill-climbing algorithm.

Interesting. Note that the backwards solver of bidicalc does not use the previous input values of variables at all.

Somebody did this back in the DOS era. The program was sometimes called "the crooked accountant's spreadsheet", because you could start with the outputs you wanted and get the input numbers adjusted to fit.

Anyone remember?

Cool!

Constraint propagation from SICP is a great reference here:

https://sicp.sourceacademy.org/chapters/3.3.5.html

I wasn't aware of this chapter, but I did use constraint propagation for the solver (among other things), thanks!

“Formulas that update backwards” is the main idea behind neural networks such as LLMs: the computation network produces a value, the error in this value is computed, and then the error quantity is pushed backward through the network; this relies on the differentiability of the function computed at each node in the network.

"Formulas that update backwards" isn't really the main idea behind neural networks. It's an efficient way of computing gradients, but there are other ways. For example forward propagation would compute a jacobian-matrix product of input wrt output with an identity matrix. Backpropagation is similar to bidi-calc to the same extent as it is similar to many other algorithms which traverse some graph backward.

I think you should be able to use bidi-calc to train a neural net, altough I haven't tried. You'd define a neural net, and then change it's random output to what you want it to output. However as I understand it, it won't find a good solution. It might find a least squares solution to the last layer, then you'd want previous layer to output something that reduces error of the last layer, but bidi-calc will no longer consider last layer at all.

All those words and you forget to provide people the breadcrumbs to learn more for themselves.

The term of interest is "backpropagation".

Won’t another breadcrumb be Prolog and “declarative programming”[1].

Wasn’t Prolog invented to formalise these kinds of problems of making the inputs match what the desired output should be.

[1] https://en.wikipedia.org/wiki/Declarative_programming

Yes, I'm glad to see a comment on Prolog. I think of it as _the_ foundational programming language for solving such problems. It isn't so much that it's a back propagation language; it's just that, based on which variables are bound at a given point, it will go forward deductively, or backwards inductively.

Prolog has basically nothing to do with calculus.

This is pretty cool.

This functionality is called ‘break back’ in a lot of enterprise modelling software. See [IBM TM1](https://www.ibm.com/docs/en/cognos-planning/10.2.1?topic=bre...) and [Anaplan](https://help.anaplan.com/breakback-1b7aa87d-aa13-49f6-8f7d-d...). They generally work in terms of scaling the inputs to match the new output though, which is a bit more basic than this approach.

Interesting, thanks! I had never heard of this. Yes, bidicalc is much more advanced. You can update any value of an arbitrary dependency graph of cells.

Ah, two way data binding. If you've used any frameworks before React (and a couple earlier ones with the same philosophy) you'll understand how it becomes a spaghetti mess over time.

Not necessarily if you're following best practices. Modern angular is quite scalable and uses two way data binding

This is great.

Since you've asked about bugs, I tried pushing the limits and found the following:

    A1: 100          B1: =100-A1
    A2: =A1*(100-A1)
    A3: =A1*B1

However, setting A3 to exactly 100 doesn't work, even though setting it to 101 or 99 (or even 100.000001) does work.

Another limit:

    A1: 100          B1: 100
    A2: =A1+B1
    A3: =A1*B1
    A4: =abs(A2-100) + abs(A3-100)

In case you can't tell from that last example, I think being able to fix the intended values of multiple outputs simultaneously would be interesting. If you were to give more details about the solver's internals, I'd be keen to hear them.

I think the concept is solid. I’ve only had a few minutes of playing with it, but I have the opinion is that from a UX perspective constants are more common than variables. So perhaps a cell containing a constant should not have a #, but a variable should.

This is really cool! It's like Excel's goal seek but can also handle the case of arbitrary input cells. Goal seeek can only handle one input and one output cell.

But how do you handle the case where multiple variables can be changed? If multiple input cells is the key difference from Goal seek, i think some more rigor should be placed into the algorithm here

e.g. setting A1 + B1 and wanting the result to be 5. Currently it bumps both A1 and B1 equally. What's the thought process behind this?

Yeah. The UI could have a lock icon to set, eg B1 should stay fixed and then only A1 would change.

It supports this. If you type # before a number, like #5, it's made constant.

Wow! See the classic https://en.wikipedia.org/wiki/TK_Solver

and Borland's Eureka solver https://nagodede.github.io/eureka/

A bidirectional formula is also known as an integrity constraint in databases (plus some triggers for restoring the constraint upon updates)!

The idea is very interesting. As a default strategy you could preserve the ratio of inputs by scaling them to match the scaling of the output, instead of making them equal (for addition). Similarly, for multiplication, you could preserve the ratio of inputs as well, by scaling them by nth root of the scaling factor of the output.

Currently the solver does not use the previous values of inputs at all when solving. But it could use it in some cases as a heuristic I guess, yes!

The examples are great and these bidirectional calculators are something that people would love to have in traditional spreadsheets.

So much so that Credit Suisse, which basically was running everything on heavily modded Excel, created a full language whose outputs were Excel spreadsheets capable of doing that. That thing called “paradise” was a total monstrosity but showed how much people wanted this.

That said, you really need a way to set which cells are fixed and which cells are allowed to move if you want to move past basic examples.

Most times you know what you want to do. like => if the user modifies that cell, find a solution for those specific ones.

If you can enter that info, then you have a lot more constrains for your solver and will avoid a lot of edge cases where everything goes to 0, and you can check that the calculation entered is indeed reversible or not, or if it could have multiple solutions, and so on.

> and these bidirectional calculators are something that people would love to have in traditional spreadsheets

People want them in general programming languages too. I can't count the number of times I had to implement the same function multiple times, but backwards in various ways.

Constants are supported, use # as a prefix, e.g.; #50.

I'd like to add more constraints in the future like a domain constraint for variables.

Could you build an inverse kinematics solver with this? (I recently watched a youtube video of someone iteratively working out the solutions for a robotic arm, by alternating modifying the inputs and the results)

That's an interesting example I hadn't thought of. Probably? I'll need to try it. Thank you for the suggestion!

I think one issue will be that trig functions are kinda weird because they are non-injective. So they work but they are awkward (try solving cos(A1) = 0.5). Inverse kinematics is so well studied, you're probably better off using a dedicated algorithm.

Closed form solutions might be more efficient (in time, energy) and perhaps more numerically stable.

Phenomenal! This is a solid prototype, clean execution. I've had exactly this idea, too: I've already given the spreadsheet the relationships between these values, why can't it just work backwards when values change? That premise hides a ton of complexity, though, I'm sure. Lots of scary matrices.

Thanks! Yes I loved the ratio of apparent simplicity to underlying complexity of this project. I have filled around 120 pages of draft and notes just for the math of the solver :)

Sympy can (often) solve under constrained systems in terms of the free variables. The problem I run into is discrete constraints that make solving less closed form and more combinatorial search. When textbook amplifier formulas significantly diverge from physical reality I model the errors as linear correction factors and use gradient descent to correct it in a few experiments, but I’m curious if there is software that has solved this problem.

Sounds a bit like Prolog with a Spreadsheet UI?

you might like https://omrelli.ug/g9/ which is a similar concept but for graphics

I do like g9! It was a strong inspiration for bidicalc actually!

Reminds me of functional logical programming languages like Verse. when you specify the output and ask for the inputs, you get all possible inputs.

Came here to say the same thing but with reference to Prolog.

Hm? I don't get it.

What's the point of calculating backwards non-invertible operations such as addition? Isn't the result just arbitrary?

I made this mostly as a fun challenge :)

You are right that there is some arbitrariness involved when picking a solution, however it's a bit more subtle than that.

Let's say our problem has N free variables.

Step 1 is finding the subset of R^N that is the solution to the root finding problem. If this subset is a point, we are done (return that point). Note that if there is no solution at all bidicalc should correctly report it.

Step 2 is if the solution subset is not a point. Then there is multiple (maybe even an infinity of) solutions, and picking one is indeed arbitrary.

does the algorithm tries to make minimal changes to the free variables ? If we have 1 + 1 = 2 and change 2 -> 4 then -100000 + 100004 = 4 is also a valid solution. When I tried it it changed it to 2 + 2 so perhaps there is optimization but also a valid optimization can be minimal free variable changes in which case it would be 1+3 = 4 and we update 1 free variable instead of 2. I have no idea which is better just curios how it works. I like the idea very much.

The actual heuristic used to pick a solution from an infinite solution subspace is a bit too complex to explain in a comment. I really need a blackboard :D The main goal was actually to find a solution, any solution at all, and fast. I wanted the backwards update to be very fast to feel as magic as possible. So the heuristic is pretty simple and could definitely be improved!

They said this:

> Even a normal spreadsheet is fairly complex beast. But the novel thing about bidicalc is the backwards solver. Mathematically, updating a spreadsheet "backward" is a (potentially underdetermined) root finding problem, because we are trying to find a vector of unknowns such that , where F is the function computed by the cells formulas, and G is the objective value entered in the cell. Note that F is not necessarily a single formula, but the result of composing an upstream graph of cells into a single function.

> The actual root-finding solver is a custom algorithm that I made. It a general purpose algorithm that will find one root of any continuous-almost-everywhere function for which a complete syntactic expression is known. It uses a mix of continuous constraint propagation on interval union arithmetic , directional Newton's method and dichotomic search. It is of course limited by floating point precision and available computation time.

But that really doesn't answer your question. I see no reason why the solver wouldn't decide every time it had a two-variable summation that ADD(X+Y) doesn't reverse to X=-90 and Y=100.

It is for fun. Commercial products do not support this because functions are generally not invertible.

I would expect the opposite.

Commercial products are run by product managers: they do whatever the business needs that day, and if it doesn't work for most inputs, "that's fine, our users will only ever need addition". Fun open source projects, run by the same programmer who does the implementation, obsess over finding the generic solution to inverting a function and end up with a version that isn't useful for anyone's specific case.

You can't even invert addition. If A+B=C and you change C, you can not infer A and B without making assumptions, like scaling them equally.

You could add hints as a feature to this. E.g. interest = rate × principle

The user hints principle is preferred fixed so they can see what rate is needed for a givem amount of interest.

Hints could have a precedence order (then prefer to fix earlier terms on an operation on a tie breaker.)

Very cool!

I'd love to see a version where cells are "torn off" and named as they were in Lotus Improv and one had a "formula pane" where one could see all the formulae for a spreadsheet.

Would it be possible to create this in Python so that it could be a part of pyspread?

LOL! Gemini suggested to implement this to me literally yesterday: bidirectional computations. The example was that given a temperature in Celsius and Fahrenheit, modifying either of them should update their counterpart. In angular that would be two linked signals for instance, but even that is a bit fringe. Gemini was going for something even more elaborated.

I told Gemini that spreadsheets were actually not doing that and that I had ways to implement that behavior without the complexity.

Just writing that to show the rabbit hole people are going to fall into if they let their llms go brrr. ;D

In any case, the problem is interesting. The point was to include bi-directionality inside a graph of computations so that we didn't get bogged down by cycles. The benefit being that it would handle float precision issues.

My more manual solution expect that floats precision issues are handled explicitly. I think that this level of explicitness is needed anyway for proper floating point error mitigation.

That's weird, Gemini told me not to do this.

To not do what, to not implement a constraint solver for bidirectional formulas? If you input my above comment it is for sure going to weigh the pros and cons. https://gemini.google.com/share/f40bf53d9c21

Excerpt from the initial convo with gemini: Thinking with 3 Pro December 11, 2025 at 09:59 PM

2. Propagators (Constraint Networks)The Problem: Your valueref binder and watch logic handle one-way data flow well. But complex forms (e.g., "Start Date must be before End Date," or "Fahrenheit <-> Celsius") require messy, cyclic event handlers to keep everything in sync.The Academic Concept: Propagators (Alexey Radul / Gerald Sussman, MIT).Instead of functions ($A \rightarrow B$), you define Constraints. A network of constraints seeks a consistent value for all connected nodes. It is "multi-directional" by design.

I see my message above being downvoted, I don't even know why ;D

In the end Gemini did agree that it was not necessary to introduce this level of complexity for my use case.

To be fair, Victor goes further because he adds a solver on top. In the research of a solution that might make sense. The issue in general is that not everything has a reverse operation so, in a sense, it is but an approximation.

A 2d sketcher with constraints is kind of similar. For example the equation

A = B + C

Where A, B, C are the lengths of 3 parallel lines. Within the sketcher you can drag the length of any one of those lines and the other two will adjust to keep the constraints.

Yes! I'd really like to make something graphical in this same idea space next. See g9.js for example, or parametric CAD software like FredCAD which kinda does what you said.

Thanks for sharing. It’s an alluring idea. I think you’d find the concept of statistical identifiability interesting.

Could this easily represent a Kalman filter and other typically complex control problems?

Makes me imagine plotting a inverted pendulum and other real time systems. Could a cell variable be set to Time?

Super cool! Well done. Now take it down and never let Microsoft get their hands on the code, or the entire economic system will go down in flames.

C-levels have been drawing random lines for years[1] - the invention of the graph let the genie out of the bag.

[1] https://x.com/gothburz/status/1999124665801880032

Excellent (sorry accidental pun)

This is a nice exploration.

interesting. like Excel Solver? or OpenSolver, Gurobi, other optimizers? or different objective?

Never used any of those, so I don't know! I'd be curious to read a comparison from anyone who knows about them.

I think what's pretty unique about the bidicalc solver that I made is that it does not depend on the previous input values to update backwards. It's truly solving the root finding problem. The advantage is that there are never any "stuck in a local optimum" problems with the solver. So you can solve difficult problems like polynomials, etc.

What do you use it for?

What a neat idea!

set A1 = 3 set B1 = 4

set C1 = A1 + B1 = 7

now change C1 = 14 expected A1 = 6 expected B1 = 8

what it did A1 = 7 B1 = 7

great

Why do you think that 6+8 is a better solution than 7+7?

It preserves an implicit relationship, ratio, between A and B.

When C1 changes from 7 to 14, that’s a scalar change. The least-assumption, information-preserving update is to scale both inputs by the same factor.

amazing

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