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[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-πŸ”Ÿ-R-Y. ...The πŸ”Ÿ 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 😏:πŸ™ƒπŸ™ƒ.

I've managed to figure out these rules.

😳 is greater than πŸ˜”, which is greater than 😎.

πŸ˜‘ times πŸ˜† equals πŸ˜‹.

πŸ˜‘ times πŸ™„ times πŸ˜‘ equals 😏.

😬 equals πŸ˜‘ times πŸ˜‘.

Anything multiplied by πŸ™ƒ is πŸ™ƒ.

πŸ™„ times πŸ™„ equals πŸ˜‚.

πŸ˜‘ times 😬 is 😠.

πŸ˜‘ times πŸ™„ equals 😘.

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

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πŸ™ƒ = 0 (because πŸ™ƒ * x = πŸ™ƒ)

Because

😬 equals πŸ˜‘ times πŸ˜‘ we know
πŸ˜‘ = 2 or 3
😬 = 4 or 9

But

πŸ˜‘ times 😬 is 😠
so 😬 must be 4 because 9 x something > 1 would be too large
This means πŸ˜‘ = 2

So

2 times πŸ™„ times 2 equals 😏. Which means πŸ™„ can be 1 or 3
And 😏 is 4 or 12. But 4 is 😬, so 😏 is 12 and πŸ™„ is 3

Thus

πŸ™„ times πŸ™„ equals πŸ˜‚. Means
πŸ˜‚ = 9

And

πŸ˜‘ times 😬 is 😠 means 😠 is 8
πŸ˜‘ times πŸ™„ equals 😘. Means 😘 = 6

Because

2 times πŸ˜† equals πŸ˜‹.
πŸ˜† must be 5 based on what’s left and πŸ˜‹ = 10

And

😳 is greater than πŸ˜”, which is greater than 😎 means
😎 = 1, πŸ˜” = 7 and 😳 = 11 because they are the only ones left

So

πŸ™ƒ = 0
😎 = 1
πŸ˜‘ = 2
πŸ™„ = 3
😬 = 4
πŸ˜† = 5
😘 = 6
πŸ˜” = 7
😠 = 8
πŸ˜‚ = 9
πŸ˜‹ = 10
😳 = 11
😏 is 12

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  • $\begingroup$ 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). $\endgroup$ – Exal Dec 15 '18 at 23:14

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