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You're a cosmonaut dedicated to exploring and understanding strange new worlds. The latest world you've arrived on has a species with three distinct genders which you've named Alpha, Beta, and Gamma. After careful observation, you determine the rules for the gender of offspring are:

  1. A Gender cannot mate with its own gender.
  2. If an Alpha and a Beta mate, they produce two Gammas.
  3. If an Alpha and a Gamma mate, they produce two Betas.
  4. If a Beta and a Gamma mate, they produce two Alphas.
  5. After mating, these strange creatures promptly die. (They can only mate once)

You start to wonder: how can this species exist? Surely given enough time, all the members of this species would be of one gender, unable to mate, and the species would die off. After pondering and observing a while, inspiration strikes. Now satisfied that this is impossible, you leave the planet in search of more interesting puzzles.

How did you know this species can't die off?

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    $\begingroup$ Is mating the only cause of death on this particular planet? $\endgroup$ – GOTO 0 Jan 22 '16 at 20:54
  • $\begingroup$ yes I should have made that more explicit $\endgroup$ – Slepz Jan 22 '16 at 20:56
  • $\begingroup$ I think a better question could have been, what is the minimum population in order to allow the species to survive, given that no matter how many of them there are, if they decide to be selfish (and mate with the wrong partner) they can effectively die off fairly quickly $\endgroup$ – Spacemonkey Jan 22 '16 at 22:00
  • $\begingroup$ that one's too easy though. The answer is 3. Two of one gender and one of another. $\endgroup$ – Slepz Jan 22 '16 at 22:08
  • $\begingroup$ I'm not sure about your definition of "impossible". $\endgroup$ – Tin Man Jan 23 '16 at 0:14
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First observation:

Since every mating produces two offspring and results in the immediate death of both parents, and since there is no other source of death on the planet, the total population of creatures is constant over time; only the relative population levels of the genders can change. (We'll leave the question of how these creatures reached their current population level for worldbuilding.SE)

Second observation:

The population naturally self-balances. That is, if all but one of the population is of gender 'alpha', and if the last other individual is of gender 'beta', then an alpha will mate with the beta, producing two gammas. And each of those will then mate with another of the alphas (since they can't mate with each other), producing four more betas. And so on, with the overbalance of alphas slowly decreasing with each cycle.

Let's do some eugenics, boys and girls!

So in order to prove what would have to happen to make the population collapse to a single gender, let's try to engage in selective breeding to make that happen intentionally. For the following discussion, assume that we want only alphas to remain.

Now, it's easy to reduce the population to only alphas if we have precisely the same number of betas as of gammas; we'd just pair the betas with the gammas and produce entirely alpha offspring, while the betas and gamma parents die, and we're done. But to do this, we need to have precisely the same number of betas as gammas, because if there is even a single surviving, unmated beta or gamma, then the population diversity will eventually recover given enough mating operations (as per the second observation, above).

So let's assume that we don't have the same number of betas as of gammas. If the number of betas doesn't match the number of gammas, can we perform selective breeding in order to make those numbers match?

Well, there are three moves we can make:

  1. Mate alpha with beta. We wind up with one fewer beta, and two more gammas.
  2. Mate alpha with gamma. We wind up with one fewer gamma, and two more betas.
  3. Mate beta with gamma. We wind up with one fewer beta and one fewer gamma.

We can eliminate move #3 as irrelevant; it doesn't bring the number of betas closer to the number of gammas. Moves #1 and #2, on the other hand, can be used to bring their numbers closer together by 3.

And therefore:

Regardless of the total population of the planet, if the difference in the number of beta individuals from the number of gamma individuals is a multiple of 3, then we can perform selective breeding in order to make the total number of betas equal to the total number of gammas. And once we've done that, then we can make the betas and gammas mate and die off, resulting in a population consisting only of alphas.

And on the other hand, if the difference between the number of betas and the number of gammas is not a multiple of 3, then we cannot ever make them equal via the available mating operations, and it's impossible for there to ever be a population consisting only of alphas.

Of course, this is also true for each of the other pairs of genders; if the difference between alphas and betas is a multiple of three, we can equalise their numbers and produce a population entirely of gammas, and if the difference between alphas and gammas is a multiple of three, we can equalise their numbers and produce a population entirely of betas.

And finally, the answer:

The puzzle specifies that we do eventually determine that it's impossible for the population to become entirely composed of any one gender.

To reach that conclusion, we must have taken a census of how many individuals there are of each gender, and determined that no pair of genders have a difference that is a multiple of three.

As a result of this, by the logic above, no mating operations can possibly occur which would result in any pair of genders having the same number of individuals. And since no two genders can ever have the same number of members, they cannot ever be bred together to remove all members of both genders at the same time.

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  • $\begingroup$ Figured out how to say this this morning, but now that I'm finally at a computer you beat me to it. This should be the correct answer. $\endgroup$ – ricksmt Jan 23 '16 at 23:42
  • $\begingroup$ I agree that this is also a better explained answer $\endgroup$ – Spacemonkey Jan 25 '16 at 14:49
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Give each gender a value: Alpha is 1, Beta is 2, Gamma is 3. Then consider what happens to the total value when two mate.

  • Alpha and Beta produce two Gamma: +3

  • Alpha and Gamma produce two Beta: +0

  • Beta and Gamma produce two Alpha: -3

The total value always remains the same mod 3. In particular, if it is not a multiple of 3, then it will never be a multiple of 3.

Now, if the number of creatures is divisible by 3, then if they were all the same gender, the total would also have to be divisible by 3. So if there are a multiple of 3 creatures, and the total value is not a multiple of 3, then the species cannot end up with only one gender.

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  • $\begingroup$ How are you arriving at the +3/0/-3? $\endgroup$ – question_asker Jan 22 '16 at 21:27
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    $\begingroup$ @carl thanks - I understood where the values were coming from, I didn't intuitively guess "replacement" as the operation $\endgroup$ – question_asker Jan 22 '16 at 21:44
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    $\begingroup$ I fail to see how this makes dying off impossible. If the population is spread evenly across all 3 genders, and that for some reason all Alphas and Betas decide to mate together, you will end up with a population of only Gammas, effectively dooming the species... $\endgroup$ – Spacemonkey Jan 22 '16 at 21:57
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    $\begingroup$ @Spacemonkey If the population is spread evenly, then the total value is a multiple of 3. $\endgroup$ – f'' Jan 22 '16 at 22:04
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    $\begingroup$ @f'' After some thought, I agree. In order to avoid the possibility of all individuals becoming the same gender, the number of one gender mod 3 must be 0, another gender mod 3 must be 1, and the third gender mod 3 must be 2. (for reasons discussed in my own answer). And with those values (the values required in order to be unable to attain a mono-gender population), we're guaranteed that the total population will be a multiple of 3. $\endgroup$ – Trevor Powell Jan 24 '16 at 11:42
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I'll go and post a not-so-lateral-thinking answer.

If individuals only die after mating then

the population never dies off, regardless of gender distribution

because

mating doesn't change the number individuals alive.

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I do not see any trick at all. Assume we had 3 pairs of parents: Alpha + Beta, Alpha + Gamma, Beta + Gamma. That is 2α + 2β + 2γ in total. The first couple gets 2γ as offsprings, the second one gets 2 bouncy baby β’s, and the third couple becomes the proud parents of 2α. In total, that is 2α + 2β + 2γ.

You say:

Surely given enough time, all the members of this species would be of one gender, unable to mate, and the species would die off.

This is completely absurd: as long as there are 6 creatures that are 2α + 2β + 2γ, they are never going to die out as long as all of their children survive because each generation, the gender composition is the same. The stationarity of the population it attained at the 1:1:1 gender ratio, and the excess guys are... well, not so lucky. Not them, but their children, of course.

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    $\begingroup$ But what if for some reason, each alpha decides to mate with a beta and ignore the gammas? Now you would have 2 old gamma who can't mate, and 4 new gammas, and that's the end of that. $\endgroup$ – Kate Gregory Jan 22 '16 at 23:57

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