# An Elementary Connect Fano Plane

I never thought I'd be able to make a connect Fano plane out of chemical elements, since there are so few to choose from. And I was right, but I got really close.

So if you haven't seen previous connect Fano planes, they are very much like a connect wall, except you are given seven words (element names in this case). Your job is to determine seven categories such that the elements and the categories form a Fano plane: each category matches exactly three elements, each element is in exactly three categories, each pair of distinct categories have one element in common, and each pair of elements lie together in one category. Ideally, if you get the categories correct, there will be a final answer that matches all seven categories, but I was only able to get six categories to work...so one of the categories will NOT match the final answer.

Obligatory Fano plane picture:

So here are the elements:

COPPER, GALLIUM, LUTETIUM, POLONIUM, PROTACTINIUM, RUTHENIUM, THALLIUM

Solver Note: I used everything I could think of to come up with categories. Element names, atomic numbers, atomic symbols, physical properties (I cap at high school chemistry, so don't pull out your CRC Handbook), and the periodic table are all fair game.

Good luck!

HINT #1:

The answers as of 1300 UTC, 17 July 2020, contain between them four of the lines I used. A line which has not appeared yet is defined by a physical property of the elements.

HINT #2:

To find another line that has not appeared yet, look at the periodic table.

• Could someone point me to a formatting cheat sheet for tabular answers? I'm having trouble determining the current LaTeX support on SE, and it looks like monospace and spoiler tags don't mix. Update: pre kinda works, but I suspect there's a better way. – chrylis -cautiouslyoptimistic- Jul 16 '20 at 15:23
• (Dear @chrylis -cautiouslyoptimistic-, i converted your answer to MathJax (La(TeX)). If you haven't seen it already, MathJax basic tutorial and quick reference at Mathematics Stack Exchange has many answers.) – humn Jul 16 '20 at 20:56

## 3 Answers

After a LOT of false trails and dead ends I believe I have FINALLY hit upon the solution!

PROTACTINIUM, LUTETIUM, GALLIUM - atomic numbers comprise only odd numbers (#91, #71, and #31, respectively);

PROTACTINIUM, COPPER, POLONIUM - sum of digits in atomic number is greater than or equal to 10 (#91 = 10, #29 = 11, and ##84 = 12, respectively);

PROTACTINIUM, THALLIUM, RUTHENIUM - density greater than 10 g/cm3 (11.72, 11.85, and 12.37, respectively);

LUTETIUM, COPPER, RUTHENIUM - second letter of symbol is 'u' (Lu, Cu, and Ru, respectively);

GALLIUM, COPPER, THALLIUM - diagonally adjacent to Cadmium (Cd) in the Periodic Table;

GALLIUM, POLONIUM, RUTHENIUM - named after countries on the European mainland (France, Poland, and Russia, respectively);

LUTETIUM, THALLIUM, POLONIUM - occupy the 6th row (period 6) of the Periodic Table.

There is an element which meets 6 of these 7 criteria, as set out by the OP, which is:

GOLD

- Atomic number is 79 - both odd digits;
- Digits of atomic number sum is greater than 10 (7+9=16);
- Density is greater than 10 g/cm3 (19.32);
- Second letter of symbol is 'u' (Au);
- Diagonally adjacent to Cadmium in the Periodic Table;
- Is not named after a country on the European mainland;
- Occupies the 6th row of the Periodic Table.

And for ease of checking some of the Periodic Table-based criteria, here's a copy of it with the relevant elements highlighted:

• That's it! All the lines are the same as mine, and you have the right final answer. Just as a point of information, two of my categories were different than yours (though not terribly so). For rot13(CEBGNPGVAVHZ, YHGRGVHZ, TNYYVHZ, zl pngrtbel jnf "ngbzvp ahzore rdhny gb 3 zbq 4", ohg lbhef vf n ovg rnfvre naq fvzcyre). For rot13(CEBGNPGVAVHZ, PBCCRE, CBYBAVHZ, zvar jnf n yvggyr rnfvre: "ryrzrag anzr pbagnvaf gur yrggre B"). Of course, seeing your categories makes me wonder if I could go back and find a...nah, that's enough of that :-) Thank you kind Stiv! – Jeremy Dover Jul 18 '20 at 20:40
• Also, ISWYDT with the periodic table...I feel like I've seen it someplace before :-) – Jeremy Dover Jul 18 '20 at 20:42
• @JeremyDover Gosh, that second one... can't believe I didn't spot that one - went through the whole alphabet (or thought I had!). I'll leave it as it is, since it all works... And yes, I'm going to reel that template out whenever possible - get my money's worth...! – Stiv Jul 18 '20 at 22:15

Edit: Not the right answer but may serve as a partial as OP has suggested some of the categories match the intended solution (I misread the question before, there is a final answer which should belong to six of the constructed categories).

Configuration

POLONIUM
GALLIUM PROTACTINIUM
THALLIUM
COPPER LUTETIUM RUTHENIUM

Groups

{COPPER, LUTETIUM, RUTHENIUM} - chemical symbols contain letter U.
{COPPER, THALLIUM, PROTACTINIUM} - chemical symbols don't match first letters of name.
{COPPER, GALLIUM, POLONIUM} - named after EU countries (Cyprus, France, Poland).
{POLONIUM, PROTACTINIUM, RUTHENIUM} - all end in NIUM.
{POLONIUM, THALLIUM, LUTETIUM} - all in period 6.
{GALLIUM, PROTACTINIUM, LUTETIUM} - atomic numbers contain two odd digits.
{GALLIUM, THALLIUM, RUTHENIUM} - all belong to groups which are Fibonacci numbers.

• I don't doubt that the lines are correct, but do you have a final answer that matches six of the seven categories? Several of your groups do match mine. – Jeremy Dover Jul 16 '20 at 15:47
• @JeremyDover Ah, I completely misread the question, sorry. Will think on this. – hexomino Jul 16 '20 at 15:50
• No worries. The first one, I wasn't even sure it was possible. For subsequent ones I've tried to make them more like Connect Walls. – Jeremy Dover Jul 16 '20 at 15:50
• @JeremyDover Just to clarify, the final answer is an element which fits in six categories, right? If so, I don't think mine works. – hexomino Jul 16 '20 at 15:58
• Yes, it is an element which fits in six of the categories. That's why this was so &\$*#ing frustrating to make :-) – Jeremy Dover Jul 16 '20 at 15:59

I've been able to come up with a number of categories. Here are three of them as a partial answer (ASCII art because I'm having trouble figuring out the current state of LaTeX tables on SE):

$$\begingroup \small \newcommand \row[8] {#1&\sf #2&\sf #3&\sf #4&\sf #5&\sf #6&\sf #7&\sf #8} \begin{array}{l|ccccccc} \row{ }{Cu}{Ga}{Ru}{Lu}{Tl}{Po}{Pa}\\ \hline \row{\textsf{named after a country}^*}{ }{ X}{ X}{ }{ }{ X}{ }\\ \row{\textsf{prime atomic number} }{ X}{ X}{ }{ X}{ }{ }{ }\\ \row{\textsf{post-transition metal} }{ }{ X}{ }{ }{ X}{ X}{ }\\ \end{array} \endgroup$$

* a recognized country at the time of the element's naming

• Wouldn't copper (Cu) fall into your second category? – bobble Jul 16 '20 at 15:28
• @bobble Sigh, yes, thanks. – chrylis -cautiouslyoptimistic- Jul 16 '20 at 15:28
• Here's a category to make up for the one I invalidated: elements with a double letter. (Cu, Ga, Tl) – bobble Jul 16 '20 at 15:32
• I think copper falls into your first category too - it's named after Cyprus, technically... – Stiv Jul 16 '20 at 15:33
• You could fix that previously invalidated category by inverting it i.e. ROT13(flzoby va erirefr nycunorgvpny beqre). – Jaap Scherphuis Jul 16 '20 at 15:34