10
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When life is finally discovered in another solar system, the government sends a bunch of pop-culture celebrities as ambassadors to meet and greet these newfound organisms.

In this new planet system, there are three planets that host life. The first is an icy planet on which live a host of intelligent life-forms composed entirely of gelatin. The second is an extremely hot planet very close to the systems sun, under the surface of which cities of insect-like organisms thrive. The third is a large asteroid on which live a group of superintelligent computers.

As you may have suspected, the celebrities don't do so well when it comes to interpreting what the natives have to say. They introduce the three alien races to each other, and the three races get along much better with each other than with the celebrities. So, after much begging and pleading, the science and mathematics community persuades the government to send a few of its top scientists and mathematicians to this new system to share information with its inhabitants. You are chosen as one of the mathematicians.

On the first planet, you manage to extract the written representations of the numbers $10$ and $24$ from some of the native mathematicians. $10$ is written

$$\Delta -\Delta$$

and $24$ is written

$$\Delta-\nabla-$$

It seems that $\Delta$, $-$, and $\nabla$ are the only symbols that they use to write numbers, and that the others are operators of some sort.

The gelatinous mathematicians also provide you with a few simple true statements in their mathematical language:

$$\Delta -\Delta'\Delta//\Delta\Delta\nabla$$ $$\Delta -\Delta,\Delta//\Delta--$$ $$\Delta -\Delta,,\Delta\nabla\nabla//\Delta\nabla$$ $$\Delta-\nabla-//\Delta''\Delta\nabla''\Delta-''\Delta\Delta$$ $$\Delta\nabla'\Delta\nabla//\Delta\nabla''\Delta\nabla//\Delta\nabla'''\Delta\nabla$$

On the second planet, you also obtain the natives' written representation of $10$ and $24$. $10$ is written

$$||\_||||$$

and $24$ is

$$||||\_|||||||$$

They also provide you with some true statements in their mathematical language:

$$||\_||||:||||||^{-}||$$ $$|\_|:|\#|$$ $$||\_|||:|\#|\#|$$ $$||||||\_|||||||:||\#||\_||||$$ $$|||^{-}||:||||^{-}|||:||||||^{-}|||||$$

On the asteroid, the supercomputers refuse to share their highly superior system of computing, but they quickly learn the number systems of the other two races and decide to help you learn them by comparing symbols from the two systems (these are hints for if you get stuck).

First, they tell you that the numbers $\Delta\nabla\nabla-$ and $|\_||||||$ represent the same number in both languages.

...

Then they tell you that the number $\Delta$ cannot be written using addition in the second system.

...

The last hint that you can persuade them to give you is that the numbers $|$, $||$, and $|||$ are expressed in the first system as $\Delta\nabla$, $\Delta-$, and $\Delta\nabla\nabla$ respectively.

Can you translate the statements given to you by the aliens and explain the workings of their number systems?

...because the celebrities sure can't.

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10
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The first system is:

balanced ternary with $\Delta$, $-$, and $\nabla$ as 1, 0, and -1 respectively. ${}'$ is $+$, $,$ is $-$, ${}''$ is $\times$, $,,$ is $\div$, and ${}'''$ is the power operator. $/\!/$ is their equality symbol.

The second is:

expressing all numbers with primes. $n$ bars represents the $n$th prime, $:$ is equality, $\_$ is addition, ${}^{-}$ is subtraction, and $\#$ is multiplication.

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  • $\begingroup$ (For some reason, it's not letting me spoiler the symbols for the first system. Not sure why.) $\endgroup$ – Deusovi Jun 27 '17 at 19:46
  • $\begingroup$ Well that was quick. I'll have to give fewer hints next time. $\endgroup$ – Frpzzd Jun 27 '17 at 19:47

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