[This puzzle is heavily based on one from video game Zero Time Dilemma.]

I have one of those weird individualist friends--to give you an idea, he spells his name H-E-N-π-R-Y. ...The π is silent, y'see.

Anyway, he's easy enough to talk to but, like most people around this generation, he insists on texting. And the thing is, he uses his own number system--it's base-13 and, furthermore, is expressed entirely in emojis. It's always difficult to settle on a time to meet when he says he's available at π:ππ.

I've managed to figure out these rules.

π³ is greater than π, which is greater than π.

π times π equals π.

π times π times π equals π.

π¬ equals π times π.

Anything multiplied by π is π.

π times π equals π.

π times π¬ is π .

π times π equals π.

From here, can you determine what emojis correspond to what numbers, 0-C (zero through 12, base ten)?

π = 0 (because π * x = π)

Because

π¬ equals π times π we know
π = 2 or 3
π¬ = 4 or 9

But

π times π¬ is π
so π¬ must be 4 because 9 x something > 1 would be too large
This means π = 2

So

2 times π times 2 equals π. Which means π can be 1 or 3
And π is 4 or 12. But 4 is π¬, so π is 12 and π is 3

Thus

π times π equals π. Means
π = 9

And

π times π¬ is π  means π  is 8
π times π equals π. Means π = 6

Because

2 times π equals π.
π must be 5 based on whatβs left and π = 10

And

π³ is greater than π, which is greater than π means
π = 1, π = 7 and π³ = 11 because they are the only ones left

So

π = 0
π = 1
π = 2
π = 3
π¬ = 4
π = 5
π = 6
π = 7
π  = 8
π = 9
π = 10
π³ = 11
π is 12

• Well done! I think you solved this faster than I did in the game (which had the advantage of better-organized clues and prime factoring in its number construction). – Exal Dec 15 '18 at 23:14