# A geometric Matrix

The first two figures are complete. The third figure is almost complete. What shape is missing in the third figure? Explain why.

Hint 1

------ stands for a word

Hint 2

Something to do with the words themselves...

Hint 3 (strong)

• Just wanted to say, the "squares" they're in are just frames so they're not part of the puzzle. Sep 9 at 17:47
• Is that same word throughout ? rot13("bphyne") ?
– Prem
Sep 12 at 19:06
• @Prem Yes, thats the correct word Sep 12 at 19:09

New theory: The missing shape is a

LINE!

Reasoning:

We need to fill in the blanks with a single word that makes sense.
It was pretty clear right away that the word was probably 'ocular' but kudos to @prem for getting 2357 to admit it and save us time considering other options.
This gives: NONOCULAR, MONOCULAR, BINOCULAR - which account for the number of circles, or "eyes".

The more difficult task was to answer the actual question - what is the missing shape?

Searching the puzzle for clues, all I found to work with were the words 'ocular' , 'geometric', and 'Matrix'.

After a few failed attempts in other directions, I started trying to count the

number of holes in capital letters (or 'eyes') for quite a while on various sequences but had no luck.

But the new hint seemed to confirm this was a possible approach (seems to show

old letters with holes).

In fact, if we use case shown in the puzzle (not changing to capitals), we get:
Nonocular - 6 letters with NO holes (no eyes): Nnculr
Monocular - 3 letters with 1 hole (one eye): ooa
Binocular - 1 letter with 2 holes (two eyes): B

A LINE is the missing shape! (shape with one side)

• Nice interpretation but this is not the intended answer. It hasn't anything to do with angles or similar. The answer is a lot less complicated. Sep 14 at 8:17
• But you're kinda on the right track with rot13(rlrf) Sep 14 at 8:21
• Yea it's not that either. Keep trying, I think you can get this if you just focus on rot13(rlfr) Sep 14 at 12:03
• @Prim3numbah Okay, revised accordingly.
– Amoz
Sep 18 at 19:59
• rot13(Vs abg n yvar, gura creuncf n pvepyr be biny). Sep 19 at 7:15

My best guess on the whole:

The 6 letter word is Ocular , giving NonOcular (0) MonOcular (1) BinOcular (2)

the image must have some arc to connect the 2 pieces like this:

Maybe some more clues will help in concluding this

• Good job for finding the correct word. But this is not the intended answer and it doesn't explain the larger shapes. What do the larger shapes correspond to? Sep 12 at 19:36

# Solution 1: (Wrong Answer, But Still Makes Sense)

An octagon

Because

The number 6 (amount of edges on a hexagon) itself has no lines of symmetry, the number 3 has one, and the number 8 has two.

# Solution 2: (Wrong Answer, But Still Makes Sense)

A Triskaidecagon (13-sided polygon) (aka Tridecagon)

Because

The word contains two i's (eyes). Also the other spelling cannot work.

# Solution 3: (Wrong Answer, But Still Makes Sense)

I think that the answer should be

Nothing or a circle should be added to the last diagram.

Because

Non means zero,
Mon is short for mono, which is a prefix for one,
Bin is short for binary, which is related to two.

This may be related to the amount of circles, and for every circle in the diagram, three sides are reduced from the shape around them. Then, the shape outside the inner circles should be a shape with zero (straight?) edges, and so there should be nothing added or a larger circle.

• You're correct that the prefix is related to the amount of rot13(pvepyrf). But not the right reasoning for the missing shape. Notice there are six small lines also after each prefix.. they are important to make sense of in order to come up with the correct answer. Sep 11 at 7:10
• I added a hint. Sep 11 at 8:12
• @Prim3numbah Could you please check my Solutions 1 and 2? (Solution 3 is old solution) Sep 15 at 7:00
• Sry but both of the solutions 1 and 2 are not correct. I'll add another hint (probably last one) a bit later today. Sep 15 at 9:24
• @Prim3numbah Could you please tell me what your time zone is? (I need to know when the hint comes out - my time zone is GMT+10 and it is 7:38pm Thursday here) Sep 15 at 9:38

I believe the third square can be completed with:

a semi-circle.

Because:

The empty dashes in each word represent 'ocular' (something I had concluded for myself independently of other posts here), giving us the words 'NONOCULAR' (meaning "no eyes"), 'MONOCULAR' ("one eye"), and 'BINOCULAR' ("two eyes").

The equivalent number of 'eyes' is represented in the centre of each square by the number of circles. So far, nothing new compared to other answers...

However, (and here is the new insight) consider a homophone of 'number of eyes', namely 'number of I's, and look at the spelling of the names of the shapes surrounding the circles:

HEXAGON - no I's;
TRIANGLE - one I.

and next in the sequence should come a simple shape with two I's in its name, meaning one possible answer is a SEMI-CIRCLE!

The reason this sequence is so short as presented is that it gets quite hard after this to continue it without resorting to increasingly obscure shapes. For instance, the first one-word shape you could use in a fourth box would be:

a 21-sided ICOSIKAIHENAGON! (Unless you dropped the one-word requirement and counted an ISOSCELES RIGHT TRIANGLE, of course!)

• Eye considered this too but there seems to be no reason hexagon was chosen vs say pentagon, and too many options for #3. I think we need more data points, there are too many ways to force an answer.
– Amoz
Sep 15 at 16:06
• @Amoz I think (if correct) the choice is just arbitrary: "a shape - one of many possibles - which satisfies [the pattern]..." I reckon the precise selections were mainly chosen for aesthetics.
– Stiv
Sep 15 at 16:12
• This is essentially the same as Cheese Cake's second solution (but with a different arbitrary choice), which was apparently not correct. Sep 15 at 16:20
• @Jaap I hadn't actually spotted that - thanks for pointing it out.
– Stiv
Sep 15 at 16:24
• Nice try but this is not the intended approach either. But the answer can arguably be correct. Keep thinking of rot13(rlrf). Sep 16 at 7:15

The pictures are the output of a discrete, recursive function:

$$f : \{1,2,3\} \mapsto \{p_1, p_2 ,p_3\}$$ $$f(1) = 6\text{-sided polytope, 0 circles}$$ $$f(x>1) = \frac{f(x-1)}{x}\text{-sided polytope,} \ x-1 \ \text{circles}$$

For every output, the prefix corresponds to the number of circles.

Thus,

$$f(3) = \frac{f(1)}{2\times3}\text{-sided polytope,} \ 2 \ \text{circles} = \frac{6}{6}\text{-sided polytope,} \ 2 \ \text{circles} = \text{1 line and 2 circles}$$

So, a line is missing.

If location is a variable, the line would cross through the centres of the circles, such that its midpoint hits the point of tangency between the circles. I reckon this, given that the circles seem to be placed in the other shapes' centers.

I had to write polytope, and not polygon, given that $$f(x<3)$$ yields two-dimensional shapes, whereas $$f(3)$$ yields a one-dimensional shape.