The Craziest Problem I’ve Had My Students Solve This Year…

 Do you let your students use calculators?

Am I going to get rick rolled?

Bubbles

Why do you want my head to explode?

Bubbles

There’s a reason I was an English major.

I could do it, but I’m way too lazy

Bubbles

I was also an English major – applause to your students, though!

On this, yes!

Maybe.

Maybe.

I’ve calculated f(g(f(h(x)))) by hand so far, it’s getting pretty big.  I’m too lazy to go on.

On the plus side, I actually remembered the process. At one point I was about to square a polynomial all wrong, just tried to apply the power of two to each element, that would have been bad. It’s been over 15 years since I had to do these kinds of calculations in college.

There are reasons as to why I’m a math major.  Love it!!

I shall pass this on to my grad student friends.

They’re a hole in my brain where all the math should be. – Harriet M Welsh

I like the Youtube link idea.  might have to adapt that, but just make the key something like:

Preposition               -> P
Proper Adjective    -> l (lowercase L)
Refelxive Pronoun -> c
Intransative Verb  -> Q

etc.

My old enemy; Math. We meet again!

No fair Hemant, it has been over 20 years since I looked at this stuff. I have no idea where to begin.

Can someone post the answer for those of us who haven’t done math in 20 years?

No dice. I refused to show my work for my calculus teacher in high school, I refuse to do it here.

Oh my, I’m an actuary out of practice! I’ve handed it out to my office to see if anyone else can get it, because I got lost about halfway through. Too many cobwebs up in my brain!

I’ll post the answers soon.

Bubbles!

Hemant, I have done far worse by hand (PDE’s = coffee overdose). This one was big, and reminded me all about the wonders of Mathematica. I solved the bubble problem by hand too. Well for 3 bubbles.

Good job getting your students ready for University .

This is awesome! Thanks for the fun, (a break from strengths and materials statically in-determinant problems), my sis suggested I look at this… wish I had you as a math teacher in high school.

Very clever problem.  69 people have already viewed your video – how many of those were in your class?

The astute student will find that you can precalculate g(h(h(g(x))) relatively easily and then apply it in one step, saving yourself some effort.

(Also: bubbles)

Haven’t done this much math in almost 2 decades. I’m sure I went about it in an incredibly long way, filled both sides of three pieces of scrap paper, but was very pleased with myself that I managed to actually solve it correctly. Naturally there were Bubbles involved.

Does this have something to do with math?  Ugh!

Does this have something to do with math?  Ugh!

That was easy but painfully annoying. You look good with glasses :3

Oh, and Bubbles o/

I just viewed it now, 17 minutes after your comment, and it still says 69 views, for whatever that’s worth.

Haven’t done composite functions in almost 20 years!  Around the last time I blew some bubbles, come to think of it…

I feel so smart now!  Thanks, Hemant.  ;)  Bubbles!

me too …

Bubbles – thank goodness for Google Spreadsheet, I wouldn’t have been able to do that arithmetic by hand, like back in my day.

Up until g(h(h(g(f(g(f(h(x)))))))) it wasn’t to bad, but that last f(x) almost took more paper then the rest combined.

It’s been over 30 for me, but I still could solve it. Amazing the stuff we retain from our childhood.

It’s been over 30 for me, but I still could solve it. Amazing the stuff we retain from our childhood.

Another good challenge would have been to award even more marks if they could do it in four function calculation “steps” or less.

Mathception.

Mathception.

Easy as bubbles… it gets ieasierif you replace the “h(x)” for “a”, “f(h(x)” for “b” and so forth at least for sanity issues on solving it by hand.

Had to remind myself how to multiply functions together, since I haven’t had to do that since high school, but finally got it. And Bubbles.

When they’re done with all their work and have swallowed all their Tylenol, you may wish to share this with them as incentive to consider the math+computer-science dual major in college:
with h as     (select -2 f, 1 p from dual) ,fh as    (select t1.f*t2.f f, t1.p + t2.p p from h t1, h t2     union all select (-1) * f f, p from h     union all select 1 f, 0 p from dual),gfh as     (select 3*f f , p from fh      union all select 1 f, 0 p from dual),fgfh as     (select h1.f*h2.f f, h1.p + h2.p p from gfh h1, gfh h2      union all select (-1) * f f, p from gfh      union all select 1 f, 0 p from dual),gfgfh as     (select 3*f f , p from fgfh      union all select 1 f, 0 p from dual),hgfgfh as     (select -2*f f,  p from gfgfh),hhgfgfh as     (select -2*f f,  p from hgfgfh),ghhgfgfh as     (select 3*f f , p from hhgfgfh      union all select 1 f, 0 p from dual)  ,fghhgfgfh as       (select t1.f*t2.f f, t1.p + t2.p p from ghhgfgfh t1, ghhgfgfh t2      union all select (-1) * f f, p from ghhgfgfh      union all select 1 f, 0 p from dual)select sum(f), p from fghhgfgfh group by porder by p asc
Results:
F,P—-[MASKED],0[MASKED],1[MASKED],2[MASKED],339320640,4[MASKED],5[MASKED],6[MASKED],7[MASKED],8

When they’re done with all their work and have swallowed all their Tylenol, you may wish to share this with them as incentive to consider the math+computer-science dual major in college:
with h as     (select -2 f, 1 p from dual) ,fh as    (select t1.f*t2.f f, t1.p + t2.p p from h t1, h t2     union all select (-1) * f f, p from h     union all select 1 f, 0 p from dual),gfh as     (select 3*f f , p from fh      union all select 1 f, 0 p from dual),fgfh as     (select h1.f*h2.f f, h1.p + h2.p p from gfh h1, gfh h2      union all select (-1) * f f, p from gfh      union all select 1 f, 0 p from dual),gfgfh as     (select 3*f f , p from fgfh      union all select 1 f, 0 p from dual),hgfgfh as     (select -2*f f,  p from gfgfh),hhgfgfh as     (select -2*f f,  p from hgfgfh),ghhgfgfh as     (select 3*f f , p from hhgfgfh      union all select 1 f, 0 p from dual)  ,fghhgfgfh as       (select t1.f*t2.f f, t1.p + t2.p p from ghhgfgfh t1, ghhgfgfh t2      union all select (-1) * f f, p from ghhgfgfh      union all select 1 f, 0 p from dual)select sum(f), p from fghhgfgfh group by porder by p asc
Results:
F,P—-[MASKED],0[MASKED],1[MASKED],2[MASKED],339320640,4[MASKED],5[MASKED],6[MASKED],7[MASKED],8

Blech, without newlines, looks like a mess….  sorry about that.

Your recall is better than mine. To borrow from StarStuff’s post…
They’re a hole in my brain where all the math should be.

Bubbles! But I cheated – I used macsyma.

I find it fascinating how often you people presented with an equation to solve will say things like “I was an English major,” but you don’t hear people presented with a book saying “no way, I majored in math.”

Took about 5 minutes with Maxima 5.22.1, which included the time it took to remember how to use the program. The secret word is “Bubbles”.

I gotta wonder how many of the (currently) 92 views are your students and how many are from the blog…

I’ll bet this won’t display properly because I’ll bet the editor here won’t let the CODE tag go through. But, for whoever said “show your work”, here you go:


(%i1) f(x):=x^2-x+1;
                                       2
(%o1)                         f(x) := x  - x + 1

(%i5) expand (%o4);


Well, what do you know? It worked.

After doing 6 hours of Diff EQ homework, I decided I just wanted to Ti-89 this sucker. Took about 15 seconds to type it in and get the answer.

Hell, for that last f(x) calculation, I think a spreadsheet is more appropriate.

I broke down and used Windows calc once it required 3 digit multiplication.

I was an English major, and seriously, that problem may as well be written in another language.

I know I could DO the problem…I just need to sit through one of Hemant’s math classes so I can understand the concept!

I scraped by with a D+ in Algebra 2 my senior year (I should add that I had straight A’s in everything else). Needless to say, I don’t have the faintest idea how to solve this problem. I don’t even have a clue what it means.

Thanks for the practice Glad I can still do stuff like this after not looking at it in 12 years.

Heh! Another macsyma user.

Heh! Another macsyma user.

Bubbles. No calculator on the last f(x) would just be merciless.  Even then I’ve got some arithmetic wrong somewhere, but not enough that I can’t get the right answer.

I’ve always hated math like this.  While I understand the principle behind it just fine, and can even see how I would code this in my head, I never understand how to get started because no-one ever shows the starting value.

What value is “x”?  You can’t solve for it using the numbers given because the “x” from the function is supposed to contain the value used within it.

Or am I supposed to assume that the lowest value in your list of number *is* the value for “x”?

Pete…

PS:  I always hated math.  :-D

Quick sanity check for answers: Every coefficient that isn’t the constant coefficient has to be even (since you started with -2x in the polynomial). 

I do have to wonder if there’s enough ways to reason out what the coefficients have to be in general without calculating them explicitly. You can also get some data out from x^2-x+1 being a cyclotomic polynomial but I doubt that they’ve discussed that in class.

So is there some way of getting most of the coefficients without brute force? The x^8 and the constant coef are easy to work out. The middle ones seem tough without doing a fair bit of heavy computation. 

Won’t say anymore to not give away the solution. 

Also: bubbles.

Ditto. This is why I went into biology instead other sciences.

Just solve the functions from inside to outside. There are some tricks to simplify things here and there, but basically that’s how it works

The numbers at the bottom are the coefficients, not values for x

It would have been some kind of torture to not be able to use a calculator on that one- but a good lesson in being orderly and careful on paper!  It still was, even with a calculator. I’m looking at my paper and it affirms the fact that math is just another cryptic language. Kinda neat, really.
Bubbles

rofl…would have been a brilliant idea

I fucking LOOOVE math! I’ll gladly do this, mostly for fun, but also to see what lies on the other side!

x here is not a number. The idea is that one has a polynomial in x. That is a formal object that has x as a place-holder which one can (if one wants) stick in a specific value. Thus for example, f(x)=  x^2-x+1 is a polynomial. I can then talk about specific solutions it has. For example, f(x)=1 has two solutions, x=0 and x=1. But I can talk about the polynomial without talking about it at a specific value of x. 

f[x_] := x^2 – x + 1;g[x_] := 3 x + 1;h[x_] := -2 x;foo = Expand[f[g[h[h[g[f[g[f[h[x]]]]]]]]]];foo1 = Reverse[CoefficientList[foo, x]];foo2 = foo1 /. {32238973 -> “p”, 1943202 -> “l”, 23089521 -> “c”, 9784324 -> “Q”, 1453032 -> “D”, 32864311 -> “m”, 1592745 -> “v”, 23794312 -> “5″, 12313562 -> “B”, 26873856 -> “8″, 34923463 -> “X”, 5943423 -> “4″, 71663616 -> “O”, 230881 -> “M”, 77332421 -> “I”, 9847525 -> “z”, 39320640 -> “6″, 2438295 -> “K”, 60466176 -> “b”, 89203452 -> U, 6437664 -> “b”, 28465924 -> “Y”, 89348931 -> “7″, 19471451 -> “J”, 18045504 -> “o”, 53781255 -> “g”, 57103424 -> “t”, 9823578 -> “R”, 3294248 -> “9″, 22582952 -> “h”, 53747712 -> “Q”, 5884023 -> “n”};”http://youtu.be/-Z” StringJoin @@ foo2

I do like Mathematica.

Blond Powerpuff GIrl!!!

“You don’t hear people presented with a book saying “no way, I majored in math.”  ”
Maybe YOU don’t. I do, and I’m pretty sick of it.

You do realize that you are evil?!
Besides, my head was aching while I was pondering your ‘problem’, until there was a ‘KAPLAAOOWW’ when I finally got it… And then my head just hurt some more

I think Hemant just nerd sniped me.

Bubbles!  I just taught one of my friends how to do composite functions, he’s a crossfit fanatic so I compared it to the training sessions he runs.  They’re kind of fun, but this one approaches evil.

Conceptually, it’s pretty simple, really: it’s just algebra, which is just like arithmetic, except you don’t know what the actual numbers are.

I’m the exact opposite; I love tedious puzzles like these.

I understand how to solve it, the problem is without a value for x I *can’t* solve it.  There is no getting the value for the function because you cannot solve for x.

For example the first function to solve for is h(x)=-2x, or stated in an other way:

sub h(VALUE)

h_val = -2 * VALUE
return h_val

end sub

Without a value for VALUE there is no way to solve for it.  It also doesn’t make sense because the first value is going to be a negative unless the value of x is negative itself, that would make the product of the function a positive which would be in line with the list of numbers.

The problem is that there are two h(x) functions nested later in the sequence. Since none of the numbers above are negative, and since the product of each iterative function corresponds to a letter for the final answer, I can’t see how you solve this since at least one of the h(x) functions must come out with a negative number (negative times a positive is a negative).

Or maybe I’m just over thinking it?

Heh. Actually, this is the problem that finally spurred me to learn enough Mathematica to solve it.
It’s astounding how much system administration you can do without having the faintest clue what the tools one’s users need do.

Okay, so its all abstract and the value of x doesn’t matter.

Except that Hemant has a list of number values for each of the equations in the sequence which implies that x *does* have a value.  But since the original value is never stated, you can’t solve for it.

Well, unless you’re a super math whiz I guess.  :-D /* LOL */

But for me this reads a lot like:

I have 2x number of bananas and 1/2y number of apples, how many apples and bananas do I have?

If I had the value of x or y I could figure this out, but since I wasn’t given either value I’m stuck.

Mathematica makes this easy as blowing Bubbles.

see

I think you’re misunderstanding the nature of the answer to the question–that is, the numbers that go in the blanks near the top of the worksheet. They’re the coefficients in the solution.

What the worksheet is looking for is a polynomial of the form Ax^8 + Bx^7 + Cx^6 + Dx^5 + Ex^4 + Fx^3 + Gx^2 + Hx + I that is equal to the composite function f(g(h(h(g(f(g(f(h(x))))))))).

 You are being asked to solve for A, B, C, D, E, F, G, and H, not x.

A through H are the numbers that go in the blanks at the top of the sheet, and then you have to match them up with the proper characters in the code provided underneath.

Whoops—in those last two paragraphs I mistakenly left out the fact that you are being asked to solve for and mach up I as well as A-H.

You’re not solving for x. Just treat x like a blank object you pass around, basically. Just like the solution to 2x + 3x, is 5x. No further than that.

You don’t need to solve it for any particular value of x

This is about finding the correct coefficients for each polynomial term. The factor before each x^… That’s what those numbers at the bottom are. Took me a minute or two to realize that though

When I was an undergraduate, biology was the science you did if you didn’t like maths, which is why I did physics and computer science.

Now I work in bioinformatics.

Justin: what Ingen said. And I can’t count how many sciencey-engineering-computing types have refused to come to the theatre with me. Especially if it’s Shakesepare. Because it’s like another language to them! (my father is a notable exception)

I’m an English major, and I actually love Math. It’s just that sometime in high school, right around the time we started trigonometry, Math stopped loving me back. 

True enough. As I said above, I do love math. Seeing this problem makes me wish I had the time to re-learn some if it!

As a physicist, I really should be able to solve this.

But as a lazy idiot….

Thanks for your help, unfortunately I still don’t get it.

I was one of those kids in High School who needed to have extra time to complete my Math exam, the only way I could figure out the tests were to write up pages and pages and pages of charts and graphs so I could picture what it is I’m supposed to figure out.

It didn’t help that I can’t do simple math in my head and wasn’t allowed to use a calculator for the tests, something as simple as 239 x 1127 / 345 required three sheets of paper and a ton of custom multiplication charts to figure out.

So I checked wikipedia and they’re saying the coefficients are all the numbers in a formula, I’m assuming you add them all together into one long number [For example: 3, 5, 12, 20 would be 351220].

So then, assuming I’m reading this right, the first two in the sequence would be something like:

f(h(x))

h(x) = -2x
f(h) = h[squared] – h + 1

So fill in the blanks:

f(h) = -2x[squared] – -2x + 1
f(h) = 4x – -2x + 1
f(h) = 6x + 1

I don’t see a six or a sixty one in the list of numbers above, so either I’m reading the link Hemant’s included incorrectly, I don’t understand coefficients, or I’m missing something really obvious.

Either way, thanks for trying to clear it up.

Pete…

PS:  Math, the bane of my existence.  I could handle advanced physics, chemistry, computers, and programming, but because I couldn’t do, or understand, simple math (it was my only “basic” class in high school) it killed my dreams of going into computer science.  :-(

Thanks for helping, see my comments above to Rieux.  Your comment certainly helped me to understand, but I don’t think I’m getting it yet.

Oh well, Hemant will show the answer in a few days.  Hopefully I’ll understand after his explanation.

you almost have it but you lost an exponent. (-2x)^2 equals 4x^2 not 4x. 

Also you can’t add coefficients of terms which are not identical.  So 4x+2x=6x is true but with an equation like 4x^2+2x you can not combine the coefficients to 6.  4x^2 + 2x just equals 4x^2 + 2x.

From there you carry on with the functions combining the coefficients of similar terms.

So if you had for example
3x^3+6x^2+21x+300+9x^3-12x^2+3x-20
you could combine the coefficients of the similar terms and come up with
12x^3-6x^2+24x+280

The trick, of course, is to use an appropriate tool.  I chose the open source (free) Maxima.  After spending 10 minutes to figure out how to use Maxima, I figured out how to enter the functions and expand it to a solution.  Bubbles.

Of course, in algebra class, that is probably cheating…  In the real world, it is just getting the job done fast.

I’m no mathematician (though my wife is); to my understanding, a coefficient is a number that’s multiplied by something else, frequently a variable.

So in the equation 12x^2 + 37y – 48x = z, the coefficients are 12, 37, and -48. Does that help?

I’ll get you started if you’d like…

1) Start from the inside out. So, first, h(x) = -2x. That’s it for step one.

2) Working our way out, we now have f(h(x)). Since h(x) = -2x, (call this A, to make things easier) then f(A) = A² – A + 1. See, in this case, A is our “x”. We just plug in the A, so we can then insert the above expression (which A is equal to). Thus, f(A) = (-2x)² – (-2x) + 1 = 4x² + 2x + 1. Call this B to make the next step easier

3) We work our way out, and see g(f(h(x)) is next. This is the same as saying g(f(A)), which is the same as g(B). g(x) = 3x + 1, so g(B) = 3B + 1. Substitute the above expression, B, and we get: g(B) = 3B + 1 = 3(4x² + 2x + 1) + 1. We multiple 3 by each coefficient (number multiplier) in front of each variable. So, 3(4x² + 2x + 1) + 1 = 3(4x²) + 3(2x) + 3(1) + 1= 12x² + 6x + 4 (since 3(1) + 1 = 3 + 1 = 4). We call this C for the next step.

And so on and so forth. I hope this helped.

I spent a large part of my morning doing this with only a basic calculator available. Looks like I made a mistake somewhere along the line, ’cause none of hte

I spent a large part of my morning trying to solve this with only a basic calculator. I must have made a mistake somewhere, ’cause the number substitution isn’t working. I’m not re-doing it to try and see where I went wrong!

Good bit of brain exercise though.

Think of it in simpler terms. For example, what is g(h(x))? First, we figure out h(x) = -2x. Call this N for simplicity’s sake.

Next, g(h(x)). Since g(x) = 3x + 1, we can plug in whatever we need to stand in for “x”, so we can do: g(N) = 3N + 1. Plugging in the expression “N” above, we get g(N) = 3N + 1 = 3(-2x) + 1. This equals -6x + 1.

Let’s say this was it. All the question is asking is, “What are the numbers we see?” So we would fill in “-6″ and “1″.

Doing this puzzle further and further, we’ll eventually get to x², then x^3, etc., and each will have its own number in front of it… for example, 5x² – 4x + 8… the numbers we’d need are 5, -4, and 8. That’s what he’s asking.

the g(h(h( part can be simplified to 12x+1 

Most of the Math/Science people I know are pretty well versed in literature and the arts as well. Maybe I just know the wrong ones.

My brain just broke.

You don’t need mathmatica to do this – a TI-89 works just fine.  I got the numbers, I just couldn’t get the website to work. 

Oops, never mind.  I got it when I capitalized the ‘Z’.  Bubbles!

Funny tricks to send them to a congrats youtube page !

But Hemant, do you think that is cool mathematics ?? I love maths for the deep insights of many problems, for the beauty of elegant proofs, for how it triggers your imagination and challenges your mind, for the paradoxes resulting from an instransigeant logic… but here it’s just boring calculations. 
I don’t know the context, but it ain’t something that would make me want to continue maths if i was a student… juste look at all the people here that are comforted in their opinion that maths are ugly & not for them.

Wish you all the best for your teachings anyway, that’s a great job !

What the hell is “Bubbles”?
What the hell is “Mathmatica”?
What the hell is “Macsyma”?

Those are rhetorical questions. I don’t want to know the answers.

But you should all know that I now feel much more stupid than before I read this post, almost as stupid as I felt during high school math classes, that feeling of being in quicksand, while others in the class were confidently striding across the surface to dry land.

I thought it was a clearly annoying problem that only the students who really understood the material would want to attempt. It’s a challenge that’s entirely solvable, which is why I like the problem. Again, it’s extra credit and no one had to do it. Plus, I feel better knowing the problems they had on their exam were more “realistic.”

That’s a really good description of… well… that feeling.

I had the same reaction as Gprano, to be honest. It is a nice problem if you like crunching numbers for the hell of it; but there’s nothing else. What would be fun is to base the calculation on some real problem, and show the real problem in the video. It wouldn’t matter if the real problem was a bit beyond high school — might even be a bonus.The kind of thing I am thinking is calculation of Runge Kutta co-effificients, or co-efficicients for a butterworth digital filter. These have a bit of the character of the problem you’ve set — crunching lots of numbers with polynomials — but you can see a real point to it as well.

Relax. We do that for other problems. It doesn’t made this one useless.

Of course its solvable (bubbles !) and i don’t really know what you can do with your students, i’m french so i don’t know well your educationnal system). But i thought it was promoting the idea that maths are mostly calculations, which isn’t very attractive.
For instance i love Project Euler (solved ~200 problems) where you let the computer do the dirty work and do yourself the funny part.

That was fun, although working it out on the scraps of hotel stationery was a bit trying. I always loved algebra, even if things like this were a little tedious. It teaches you to pay close attention to what you’re doing.

Thanks Hemant (Mr. Bubbles?). I love a good math obsession now and then, and although I haven’t done any advanced algebra for about thirty years, your problem has me obsessing all morning. It felt great!   P.S. I also love Shakespeare.

Of course it isn’t useless; nor did I suggest it was. The idea of generating the URL to a video was clever and it was a fun challenge in number crunching.

I’m making a suggestion for sources of similar problems which have an added dimension to them. I still commend to you looking into the digital butterworth filter and the Runge Kutta
co-efficients as ways to get some more number crunching exercises but with an
extra dimension of reality to them.

I didn’t mean to come of as saying it was a bad idea or useless! I don’t think that is a fair reading of my suggestion. It’s great that you are talking about education, and I hope you keep trying out new and interesting ideas. Educators sharing ideas and suggestions is really positive.

Heh. Nope, you apparently know the right ones!

Ack!. Ive gotten as far as to make out 2 coefficients but everything else is a mess. I’ll try it again tomorrow after my morning coffee.

Awesome fun! Took about an hour to solve using only pen and paper, but there’s no way it’ll fit on just one separate sheet of paper if I have to show all my work! I think I used 6 sheets of notepad paper, plus some scraps for the longhand multiplications.

Awesome fun! Took about an hour to solve using only pen and paper, but there’s no way it’ll fit on just one separate sheet of paper if I have to show all my work! I think I used 6 sheets of notepad paper, plus some scraps for the longhand multiplications.

I know that feeling all too well! This thread reminds me of sitting in Algebra class while the teacher droned on and on about something I couldn’t understand. She might as well have been writing Chinese characters on the board as far as I was concerned.

I tried this first after not having compost functions in around fifteen years. I made the mistake of just multiplying out each of the functions which of course is not how you do composite functions. It didn’t take me long to figure out I was doing something wrong as my X’s were going to start at the thirteenth power and my numbers were going to be way smaller than the ones available. So I looked up the first tutorial on composite functions I could find and refreshed myself and started again. As a life long math dyslexic I was quite proud when I got it right. I only used a calculator for the multiplication of large numbers. The first time I took algebra 2 I failed it so this was a big victory for me. Thank you to my second algebra 2 teacher for showing me that my problems with math weren’t simply that I always got the answer wrong but I just mixed up numbers because I needed to slow down. Bubbles.

I don’t see the point in this. This is not anything that really requires any intelligence or creativity, like any good math puzzle, it’s just an overly tedious chore.

Took about 40 minutes.  Sadly I think I am better at it now then when I learned this stuff in high school. 

PS – After all that work the students deserve to see you a little more excited in the video. lol

I cheated on this one and wrote an Objective-C program to do the work for me although I did everything but that final function manually as well to double-check my algorithm. That final [diabolical] f(x) uncovered a bug but once squashed, it was nothing but bubbles.