a family affair

[BMAD]

Six unrelated beings randomly selects their mates (three female and three male). Where 1 female mates with 1 male. The result of such mating produces two offspring, one male and one female from each mating. Once the new generation comes of age, the random mating event happens again except blood relatives cannot mate. the process repeats. The beings die upon surving through the production of the second group of offspring.

How many generations until the creatures can no longer mate? If they can always choose a partner, why?

Assume that all creatures will choose a mating situation, if one exists, where everyone mates.

Edited July 27, 2013 by BMAD

[araver]

Questions:

5. Do you define blood relatives as having one common ancestor (male or female)?

Males: A, B, C. Females: 1,2,3

First generation brings up 3 males: A1, B2, C3 and 3 females: 1A, 2B, 3C (simple notation: all males start with a letter and all females with a digit).

Then 1A cannot mate with A and A1 because she has A in common.

And if 1A mates with B, two offsprings are produced: 1AB and B1A. Neither of which can mate with anyone who has A or B or 1 in their system?

If so, then the cap is the number of blood types/colors and there appears to be no infinite solution.

Just a question of when will the process block. At most 5 generations since:

1. After the second mating A,B,C,1,2,3 die. Since mating adds the number of colors offsprings receive, bicolored offsprings can only be produced during the 1st and at most 2nd mating.

2. Everyone involved in the third mating would have at least 2 colors hence each of their offspring will have at least 4 colors.

3. After a fourth mating all possible bicolored offsprings of A,B,C will die, leaving offsprings with 4 or more colors unable to mate with anyone in a fifth mating. All offsprings with exactly 3 colors (if any) may mate with their negative colors but they create 6 colored offsprings.

4. All offsprings with exactly 3 colors (if any) cannot survive after 5th mating since they were born in 3rd mating.

Trying to get an instance with exactly 5 matings is subject to the next question.

6. When you say that all creatures will choose a mating situation, if one exists, where everyone mates, does this only apply to choosing a global optimal solution? Or do they need to "choose" a situation where as many as possible mate? (Forced / Greedy approach).

If the assumption only applies to the global optimal - choose a scenario where all mate, but if at least one can't mate, choose any scenario possible including leaving aside potential mating pairs, then it may be possible to prolong the number of generations. Sorta like a "don't mate, don't die" syndrome.

Which is why I'm assuming you want to assume that "random" also selects the largest number of matings possible.

In which case, trying to prolong the lives of these creatures and dillute the blood as much as possible I can only find 4 matings possible:

Mating #1: A+1, B+2, C+3

Remaining 12: 1 life - A,B,C,1,2,3, 2 lives - A1,B2,C3,1A,2B,3C.

Mating #2 (trying to produce 2 colored offsprings): A+2, B+3, C+1, A1+2B, B2+3C, C3+1A

Remaining 6*2+6 (forefathers die) = 18:

1 life - A1,B2,C3,1A,2B,3C

2 lives - A2, B3, C1, 2A,3B,1C, 12AB, AB12, 23BC, BC23, 13AC, AC13

Mating #3

A1+23BC, B2 + 13AC, C3+AB12, 1A+BC23, 2B+AC13, 3C+12AB - all produce 6*2 colored offsprings who can't mate.

A2+3B, B3+1C, C1+2A - 6 4-colored ones

Remaining 6*2 + 6 + 6*2 = 30

1 life - A2, B3, C1, 2A, 3B, 1C, 12AB, AB12, 23BC, BC23, 13AC, AC13

2 lives - AB23, BC13, AC12, 23AB, 13BC, 12AC, and a bunch of 123ABC and ABC123s who can't mate.

Mating #4

From 18 viable candidates, only the 2-colored once can mate again before they die.

All have >4 colors, there's no one to mate with for a fifth mating.

[bonanova]

1. Shall we assume that mates are always of the same generation?

   That is, would a man be prohibited from mating with his mother-in-law?

   

2. Does the OP mean to say that once a person becomes a grandparent, then s/he dies? Wondering why this is necessary, unless cross-generational marriages could happen.

3. When children come of age, may their parents mate (again), with a different opposite-sex person from their own generation, and bear more children? Or does each person parent two and only two offspring?

4. Shall we assume that the blood relationship prohibition for marriage is absolute?

   Often, by civil law, second cousins, but not first cousins, may marry.

[gavinksong]

Three? I just went through one simulation and found that the fourth generation couldn't produce any more offspring. I could be wrong. Is it always the same, regardless of how they mate? Will think more about this after dinner.

[BMAD]

• Shall we assume that mates are always of the same generation?

  Any generation can mate with any other generation as long as they aren't related.

  That is, would a man be prohibited from mating with his mother-in-law?

  

• Does the OP mean to say that once a person becomes a grandparent, then s/he dies? Wondering why this is necessary, unless cross-generational marriages could happen.

  Once grand kids are born or ones second child is born you die (helps with keeping the numbers manageable)

• When children come of age, may their parents mate (again), with a different opposite-sex person from their own generation, and bear more children? Or does each person parent two and only two offspring?

  There is no assumption of monogamy as long as they aren't related they can mate. Remember mating is random.

• Shall we assume that the blood relationship prohibition for marriage is absolute?

  Yes. One drop of common blood prevents mating.

  Often, by civil law, second cousins, but not first cousins, may marry.

[BMAD]

Greedy (as many mate in a single instance as possible). Blood relatives means that they have a genetic relation.

[gavinksong]

I assumed the greedy approach and got a similar result (but not quite the same) as araver after running one simulation.

I used different matings however. Perhaps the result is different depending on how you pair up the individuals?

I used O and X to represent the different genders, followed by numbers indicating which of the original six they share blood with.

The way I paired up the individuals is simple and somewhat methodical. I have the old generation always pair up the same way they did in the previous mating and the new generation always pair up amongst themselves (and there is pretty much always only one way to pair them up such that all of them are paired).

Mating 1

Gen 1: O(1)-X(2) O(3)-X(4) O(5)-X(6)

Offspring

O(12) X(12) O(34) X(34) O(56) X(56)

Mating 2

Gen 1: O(1)-X(2) O(3)-X(4) O(5)-X(6)

Gen 2: O(12)-X(34) O(34)-X(56) O(56)-X(12)

Offspring

O(12) X(12) O(34) X(34) O(56) X(56)

O(1234) X(1234) O(3456) X(3456) O(1256) X(1256)

Mating 3

Gen 2: O(12)-X(34) O(34)-X(56) O(56)-X(12)

Gen 3: O(1234)-X(56) O(3456)-X(12) O(1256)-X(34)

O(56)-X(1234) O(12)-X(3456) O(34)-X(1256)

Offspring

O(1234) X(1234) O(3456) X(3456) O(1256) X(1256)

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

At this point, no more pairs can be matched.

Edited July 28, 2013 by gavinksong

[gavinksong]

I realized that I made a mistake. There is one more mating that is possible since the old generation is still alive.

Which means I got the same result as araver even though I paired the individuals up differently. So this must be the solution?

I used O and X to represent the different genders, followed by numbers indicating which of the original six they share blood with.

The way I paired up the individuals is simple and somewhat methodical. I have the old generation always pair up the same way they did in the previous mating and the new generation always pair up amongst themselves (and there is pretty much always only one way to pair them up such that all of them are paired).

Mating 1

Gen 1: O(1)-X(2) O(3)-X(4) O(5)-X(6)

Offspring

O(12) X(12) O(34) X(34) O(56) X(56)

Mating 2

Gen 1: O(1)-X(2) O(3)-X(4) O(5)-X(6)

Gen 2: O(12)-X(34) O(34)-X(56) O(56)-X(12)

Offspring

O(12) X(12) O(34) X(34) O(56) X(56)

O(1234) X(1234) O(3456) X(3456) O(1256) X(1256)

Mating 3

Gen 2: O(12)-X(34) O(34)-X(56) O(56)-X(12)

Gen 3: O(1234)-X(56) O(3456)-X(12) O(1256)-X(34)

O(56)-X(1234) O(12)-X(3456) O(34)-X(1256)

Offspring

O(1234) X(1234) O(3456) X(3456) O(1256) X(1256)

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

Mating 4

Gen 3: O(1234)-X(56) O(3456)-X(12) O(1256)-X(34)

O(56)-X(1234) O(12)-X(3456) O(34)-X(1256)

Gen 4: There are no matings possible.

Offspring

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

O(123456) X(123456) O(123456) X(123456) O(123456) X(123456)

At this point, there are no more matings possible.

Edited July 28, 2013 by gavinksong