Problem solving

One of the junior high teachers came around a while ago asking if any of us had a good “algorithm” for solving problems. Something which could, with minor tweaking, be allied to things as carried as calculus or fixing your sprinkler system.

He’s the technology teacher, so he’s got some freedom in setting up his curriculum - and he mostly teaches 7th graders, so wants to try and set them up well.

Does anyone here have any ideas for such an “algorithm”?

Not too long ago I learned the algorithm to calculate the square root of a number. I thought that was pretty cool.

Using algorithms to solve for integrals seems fun.

Binary search, N Queen problem using recursion, etc.

I have nothing useful to add, so I will take up space by adding something that is not useful, and entertaining perhaps only to me:


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Off the top of my head, this is a basic problem solving algorithm

  1. Assess the situation
  • What is the problem - scope / time constraint / importance
  • What are our resources - materials / tools / skills
  1. What is the desired result
  • Are we looking for a permanent fix or a patch
  • What are we trying to avoid.
  1. Plan and commit to a procedure
  • Steps 1 & 2 give you the guidance to proceed
  • If step 1 & 2 don’t give you the guidance, determine what you are lacking and resolve it.
  1. Assess progress / Adjust as necessary until complete

I don’t think there is really any such general algorithm.

For example, nobody has been able to turn modern science into an algorithm.

And if there were such an algorithm, it would have to depend so much on the environmental input that it would be impossible to understand it abstractly. As an example, maxwell’s equations are four relatively simple equations that lead to an incredibly diverse range of behavior. But because of this very fact, looking at them by themselves is not very enlightening in many ways.

Need a step: “Google how to solve the problem.”
Or, “Google to search for an expert on the subject in your area to solve the problem.”

Also, to the OP: figuring out the algorithm for solving the problem is essentially Step 1 to solve the problem. Tell your teacher friend this. I like “f.k.d.”'s general problem solving algorithm template, but each step needs filled in for the specific problem before tackling the problem.

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Also, consider asking chatgpt.

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But there is:

George Pólya’s How to Solve It - Wikipedia

Here is a quick summary (with “business world” addition in parentheses):

  1. Understand the (real) problem–might need to translate what is requested into what is the real problem to address

  2. Devise a plan
    a. Understand what you do know
    b. Clarify/determine what you don’t know (and need to know) that relates to the problem to be solved
    c. Identify ways to connect what you do know to what you don’t know (you might find more than one path for this)

  3. Execute (one or more of) your plans

  4. Examine the result(s)
    a. Do they address the original problem?
    b. Are the results reasonable?
    c. Should more work be done? (if so; revisit “step 2”)

It might be worth noting that much of “education” is building the toolbox for working in Step 2 (and successful execution in Step 3).

And some of working in Step 2 is exploration of what you do know to see what might be a reasonable path to fully execute. For example, might do some “trial and error” to see what happens. This might lead to a better idea of which other approaches to try (or to exclude).

An algorithm is supposed to fully determine the outcome.

That is vague enough that it looks to me more like a toolbox of possible approaches than an algorithm.

I think the Actuary Circle of Solving (is that a round (heh) anymore?) also does the trick, though most regular-world problems don’t involve gathering data.

I think a lot depends on the context of the use of the term:
Algorithm Definition & Meaning - Merriam-Webster

Starts with
“This is my problem how?”


That is true. Similar to terms like “parameter” and “space”, algorithm has migrated into more general use, and in that context loses a lot of its specificity and, in my opinion, meaningfulness.

In that case, I’d rather use a word like “guidelines.” There are certainly guidelines for solving a wide variety of problems. But then we seem to lose the ability to objectively determine whether the guidelines really “work.”

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Like “The Pirates’ Code”!

Now THAT’s the life for me!

Can’t arrrrgue with that.

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Im taking a course on problem solving right now. It seems to be mostly basic techniques and general principles then lots of practice problems and hoping for the best.

I.e. Some problems work well with diagrams. Others trying test cases then looking for a pattern. Problems where the solution required isnt in the same units as whats given, but relationships exist, use algebra.

Heres an introduction on the topic: CEMC's Open Courseware - Problem Solving and Mathematical Discovery

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When I took the Putnam, most were problems I didn’t even know how to start. Fortunately that was almost never the case on actuarial exams.

what was the solution last year?