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Here’s a diagram that looks like 12.

diagram that looks like 12

Can you dissect the red part of the diagram into n congruent pieces for every n that divides 12?

All the angles are multiples of 45° and if two lengths seem equal, they are equal.


Update:

I originally thought that I had an answer for n = 4, but I was mistaken. So I’m turning n = 4 into a bonus question. I’ll still wait for someone to prove or disprove n = 4, but it’s not essential for your answer to be accepted.

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    $\begingroup$ I’m struggling on n=4… $\endgroup$
    – RDK
    Commented yesterday
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    $\begingroup$ @RDK. Please see the update. I thought I had an answer for n = 4 but I guess I shouldn’t have posted while I was sleepy. Sorry about this. $\endgroup$
    – Pranay
    Commented yesterday

2 Answers 2

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Start with 6, bonus question not resolved:

enter image description here

Stick together parts for 3 and 2:

enter image description here enter image description here

Divide in half for 12:

enter image description here

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Seeing the update, I decided to disprove n=4. I don’t know if this proof is valid or strong…

As @z100 posted, here are the solutions except n=4.

All of the solutions

Purple dissection: n=2

Blue dissection: n=3

Orange dissection: n=6

Black dissection: n=12.

These dissection have an important property:

Except n=12 (which is a trivial congruent dissection from case n=6), they have a center of parts (Noted as yellow point below).

Center

Assuming that

there are no trivial congruent dissection of the partial shape in n=2 (like as in n=6 to n=12),

n=4 dissection should cut the outer line as below:

n=4 dissection - outer line contradiction

Then the outer line doesn’t fit a congruent condition, so it’s a contradiction.

However this (dis)proof isn’t perfect, as

We can’t trivially say that there are no any trivial congruent dissection of the partial shape in n=2 (like as in n=6 to n=12).

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