I don't know how well this puzzle will translate to a toot.

[golgaloth@writing.exchange]

@futurebird You said order. I'm a writer.

[futurebird@sauropods.win]

@gaditb

Two cards that could be flipped where the last two the students fit into their pattern.

They decided that having a repeat made no sense.

[heliograph@mastodon.au]

@golgaloth you win the writers' internet today @futurebird

[sjb@mstdn.io]

@futurebird
□
□□
□□□
□□■
□□▣
□□▷
□■
□▣
□▷
□▷□
□▷■
□▷▣
□▷▷
■
■□
■■
■▣
■▷
▣
▣□
▣■
▣▣
▣▷
▷

[mason@partychickens.net]

@futurebird

I don't see a way to determine if ■ is two or three, and ▣ is whichever of those that ■ is not. ▷ is zero. □ is one. Thus, depending on how you assign the two filled-in squares, the order is either

▷
□
■
▣
□ ▷
□ □
□ ■
□ ▣
■ ▷
■ □
■ ■
■ ▣
▣ ▷
▣ □
▣ ■
▣ ▣
□ ▷ ▷
□ ▷ □
□ ▷ ■
□ ▷ ▣
□ □ ▷
□ □ □
□ □ ■
□ □ ▣

or

▷
□
▣
■
□ ▷
□ □
□ ▣
□ ■
▣ ▷
▣ □
▣ ▣
▣ ■
■ ▷
■ □
■ ▣
■ ■
□ ▷ ▷
□ ▷ □
□ ▷ ▣
□ ▷ ■
□ □ ▷
□ □ □
□ □ ▣
□ □ ■

I suspect the ordering of ■ and ▣ is arbitrary given that □ is definitely 1, which makes them all somewhat out of order if we consider the unicode values.

▷ 25b7
□ 25a1
■ 25a0
▣ 25a3

I like patterns. Thank you again.

[stveje@mstdn.social]

@futurebird ▷ 001
□ 002
▣ 003
■ 004
□▷ 021
□□ 022
□▣ 023
□■ 024
▣▷ 031
▣□ 032
▣▣ 033
▣■ 034
■▷ 041
■□ 042
■▣ 043
■■ 044
□▷▷ 211
□▷□ 212
□▷▣ 213
□▷■ 214
□□▷ 221
□□□ 222
□□▣ 223
□□■ 224

[ingalovinde@embracing.space]

@futurebird I'm an adult, and this looks like numbers to me. There are only four letters and too many combinations made of these letters, so it's unlikely that they are words rather than numbers in base-4. (Words written with alphabet typically have much more redundant encoding with much fewer allowed combinations)
There are four single-character numbers and 12 two-character, with all possible combinations of the characters except the leading triangle in two-character numbers, no language behaves like that.
None of the numbers (besides single-character triangle) start with triangle, so I'd guess that's 0 and they're written in the same direction as our Arabic numerals. And another guess is that the list is a contiguous chunk of numbers (because at least it contains all numbers from 0 to 15)
All three-digit numbers start with empty square, so I'd guess that's 1.
All three-digit numbers start with either 10 or 11, further confirming the "contiguous range" hypothesis. There are all four possible combinations starting with 10, but only three starting with 11, so they must be 110, 111 (we already know how these look like) and 112. Therefore, nested squares is 2.

Final answer:
* It's a base-4 alphabet, encoding all integers from 0 to 112 (22 in decimal)
* Triangle is 0
* Empty square is 1
* Empty square with nested filled square is 2
* Filled square is 3

[clayfoot@mastodon.social]

@futurebird

I think assignment of ▣ and ■ is arbitrary, so I'm going to choose 2 and 3 just to make it easier to remember:
0 -> empty -> □
1 -> partial -> ▣
2 -> filled -> ■
3 -> triangle -> ▷
I can then sort the cards easily
0 to 23 in base 10, or
0 to 113 in base 4, or
□ to ▣▣▷ with the cards

[futurebird@sauropods.win]

Here is the real puzzle. l

[gemini6ice@hachyderm.io]

@futurebird linear ordering?

[ronjeffries@mastodon.social]

@futurebird two of these things are not like the others

[braxa26@mindly.social]

@futurebird The puzzle would probably take me hours to solve; I'll just admire the giant stag beetle (I think?) for a while. 😁

[anselmschueler@ieji.de]

@futurebird Assuming you're using ▷ to represent the red triangle, it seems to me like there's a card that appears twice in the image but not in your list. Namely the one at the top and at the bottom right, which I'd render as ▷■. At the top the triangle is rotated differently so it's perhaps ■▷? Still, I can't find both of those in the textual representation

[anselmschueler@ieji.de]

@futurebird I haven't checked exhaustively, but it seems to me that interpreting this as base 4 is sensible, and you can deduce that ▷=0 and □=1, but I don't see a way to pin down the values of ▣■…

[voiceofduum@mindly.social]

@futurebird I am confident the beetle will solve it

[jgrg@mstdn.science]

@futurebird

1 □
2 ■
3 ▣
4 ▷
11 □□
12 □■
13 □▣
14 □▷
21 ■□
22 ■■
23 ■▣
24 ■▷
31 ▣□
32 ▣■
33 ▣▣
34 ▣▷
111 □□□
112 □□■
113 □□▣
114 □□▷
141 □▷□
142 □▷■
143 □▷▣
144 □▷▷

[jgrg@mstdn.science]

@futurebird Digits fixed, now that I've identified the zero, and it is base 4:

0 ▷
1 □
2 ▣
3 ■
10 □▷
11 □□
12 □▣
13 □■
20 ▣▷
21 ▣□
22 ▣▣
23 ▣■
30 ■▷
31 ■□
32 ■▣
33 ■■
100 □▷▷
101 □▷□
102 □▷▣
103 □▷■
110 □□▷
111 □□□
112 □□▣
113 □□■

[futurebird@sauropods.win]

@VoiceofDuum

He's very frustrated that it takes so long to move each card to a new location.

[jgrg@mstdn.science]

@futurebird If you were to remove the zero card and add a 120 card, then the order would be unambiguous. The symbols for 2 and 3 are currently interchangeable, leaving two solutions.

[ingalovinde@embracing.space]

@futurebird once again we have four characters, four cards with each of the characters, and 12 cards with two-character combinations, so I'm just going to assume it's like the last time, base-4.
On 12 two-character cards, we get one with two triangles, one with two filled circles, one with two squares, but none with two empty circles. Therefore empty circle is zero.
Now from the card with empty circle and triangle we can deduce that if we write digits in the same order as usual, more senior to the left, less senior to the right, then the correct orientation of triangle is the one where the right angle is to the bottom left of diagonal.
Rotating all cards with triangles to their right orientation, we see that in three-digit numbers, triangles only occur in the rightmost position, and in the leftmost it's always filled circle. This probably means that filled circle means 1.
"Filled circle, square, empty circle" card is oriented properly because empty circle is zero. But we never see a three-digit number with triangle in the middle. Assuming, like in the previous puzzle, that the numbers are consequential, we get that square is 2 and triangle is 3.

The numbers on the cards are, in base-4 with Arabic digits:
32 in the second (only) column of the top row.
31, 20, 11, 2.
113, 103, 100, 13.
0, 3, 102, 12 or 21 (impossible to determine if it's upside down or not).
120, 101, 30, 22.
112, 110, 33, 1.
111, 21 or 12, 23, 10.