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My old science teacher Dr Aphorism was brilliant – not just at teaching (and, of course, science), but also at providing invaluable life advice. I always remember one day when I went to her office to talk about a coursework experiment that was not going well. She sat me down, looked at my equations, and pointed at a term involving a square root.

"I know that's the part that's wrong," I said, "but I’ve tried everything, and it just doesn’t work out."

With that, Dr Aphorism smiled and directed my attention to a poster on her wall showing a modified Periodic Table.

"I’m going to leave you here for ten minutes," she said kindly, "and I want you to focus really hard on working out the meaning behind this poster. When I come back I’m sure you’ll have a better understanding..."

TASK: What is the hidden meaning behind this poster? The (perhaps ironic) correct answer is an 8-word phrase which can be enumerated as (5, ?, 7, ?, 4, ?, 3, ?), where some word lengths have been left unknown…

enter image description here A colour guide for colour-blind solvers is available here.

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    $\begingroup$ Nice attention to detail with the thumbtacks in the corners. $\endgroup$ – msh210 Jun 8 at 12:51
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    $\begingroup$ I have to get to work (an hour later than normal...thanks Stiv :-), but I noticed rot13(sbe rirel funqrq ryrzrag, gur nygrerq ngbzvp ahzore ersref gb na ryrzrag jubfr ngbzvp flzoby funerf n yrggre jvgu gur ngbzvp flzoby va gur funqrq fdhner. V gevrq uvggvat gur erfhygvat yrggref (obgu funerq naq abg-funerq, jvgu frireny pbzovangvbaf gurerbs) jvgu inevbhf Pnrfne naq Ivtarer pvcuref, ohg ab yhpx lrg.) $\endgroup$ – Jeremy Dover Jun 8 at 14:14
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    $\begingroup$ So I know the phrase is rot13(bsgra n ceboyrz unf zber guna bar fbyhgvba), but I'm not sure yet of the logic to get there. Hoping the wife doesn't make me start painting the bedroom before I finish. $\endgroup$ – Jeremy Dover Jun 9 at 0:04
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    $\begingroup$ @Stiv: Even thought I didn't get the checkmark, I absolutely loved this puzzle, and really appreciate the craft you put into it. Wish I could upvote multiple times :-) $\endgroup$ – Jeremy Dover Jun 9 at 20:06
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    $\begingroup$ @JeremyDover Thanks, I'm glad to hear it :) It was born out of a sudden idea of "would this be possible...?" so I spent some time working out all the possible combinations that could be used, then had to devise a sentence that could be built out of them - and I was very chuffed it all worked out! Making the image in Excel was fun - plus now I have a template for future chemistry puzzles! :) $\endgroup$ – Stiv Jun 9 at 20:17
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First, notice that:

The numbers of colored elements are a bit off.

But if we take a look further:

The numbers are denoting other elements, which surprisingly share a common letter with the colored ones!

For example, for the RED one:

- Ni is numbered 22, denoting Ti.
- Ir is numbered 87, denoting Fr.
- Rg is numbered 118, denoting Og.
- Cf is numbered 58, denoting Ce.

But, cleverly, we don't stop here! Taking a look far further:

The numbers apparently share a common digit too!

For example, for the RED one:

- Ni is numbered 22, it should be 28.
- Ir is numbered 87, it should be 77.
- Rg is numbered 118, it should be 111.
- Cf is numbered 58, it should be 98.

So if we tabulate the results, and taking the "differences":

enter image description here

The last thing is to extract the final answer:

For each group, sort the digit differences, then pick the letter differences.

The final answer is:

OFTEN A PROBLEM HAS MORE THAN ONE SOLUTION.

Final remark (helped by the author):

So the wise word from Dr Aphorism to be "Remember that often a problem has more than one solution". If the narrator insists to correct the single mistake in his/her square root part, it indeed may be hard to solve or even impossible: instead, try to look at another angle!

Perhaps there are more than a single mistake here? Perhaps there is another way to solve the problem? @Johnson notices that square root may result not only positive but also negative solution (as many people forget!)

As illustrated in the poster (i.e. "there are not 1 but 2 mistakes" or "there are are not 1 but 2 solutions") surely Dr Aphorism knows how to teach her students a way of life! :)

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    $\begingroup$ @JeremyDover whoops, sorry if I'm sniping, I just realized that you've already covered most of these in the comments >< $\endgroup$ – athin Jun 9 at 4:42
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    $\begingroup$ @athin: No sniping at all...you got it all the way there, and I didn't. I like to think I can appreciate your solution all the more because I know most of what went into it first-hand :-) Great work! $\endgroup$ – Jeremy Dover Jun 9 at 11:20
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    $\begingroup$ The final answer, combined with the square root, immediately made me think of the fact that rot13(Nal cbfvgvir ahzore k jvyy unir gjb fdhner ebbgf - (cbfvgvir naq artngvir). Gur cbfvgvir ebbg vf gur bar jr hfhnyyl guvax bs, ohg creuncf gur cebs vf erzvaqvat gur fghqrag gung nabgure bar lrg rkvfgf.) $\endgroup$ – Johnson Jun 10 at 2:23
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    $\begingroup$ @Johnson dang that's legit! Adding that :) $\endgroup$ – athin Jun 10 at 2:28
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    $\begingroup$ To make it more precise, a square root always results in a positive number. However, when it is part of a solution to a quadratic equation, there could be two solutions: one with is the square root,and another is the negative of the square root. $\endgroup$ – justhalf Jun 10 at 4:04

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