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Inherited Veracity (Posted on 2012-02-09) Difficulty: 5 of 5
The land of Kivel is populated by 4 types of people:

Knights: always tell the truth;
Liars: always tell lies;
Knaves: each of a knave's statements strictly alternate truth and lies;
Transposers: a transposer's statements are always of opposite truth-value to the
  previous statement most recently made by anyone, including himself or herself.
  If no one has made a statement yet, a transposer will speak truth or a lie
  randomly for the first statement.

Scientists have discovered that in Kivelians, veracity is an inherited characteristic based on blood type. Any individual has one of six possible genotypes (KK, KO, LL, LO, KL, and OO) that produce one of four possible phenotypes:

Knight (produced by KK homozygous and KO heterozygous genotypes),
Liar (produced by LL homozygous and LO heterozygous genotypes),
Knave (produced by the KL heterozygous genotype), and
Transposer (produced by the OO homozygous genotype).

If April, Baxter, Celeste, and Douglass, are a knight, a liar, a knave, and a transposer, in some order; and if each one is a parent or child to at least one other person in the group, can you determine each person's type, and relation to the others, given the statements below?

April: I am heterozygous.
Baxter: I am a knight.
Celeste: Douglass is April's father.
Douglass: I am heterozygous.

Baxter: Douglass is not my father.
April: Baxter is a knave.
Baxter: Celeste is April's mother.
Celeste: The liar among us is homozygous.

Douglass: Baxter is my son.
Celeste: Baxter is my son.
April: There are exactly two heterozygous and two homozygous among us.
Baxter: The liar among us is heterozygous.

Celeste: April is heterozygous.
Douglass: I am related by blood to exactly two of the others here.
April: Douglass is heterozygous.
Douglass: The knight among us is homozygous.

Notes: You may find it helpful to use Punnett Squares to test which types of parents can produce which types of children. Here's an example of a Punnett Square using blood types.

Also, the spacing is given only for readability. You may assume the comments are all said in order, with no breaks between them.

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Solution Solution and working | Comment 3 of 4 |

April and Douglas both claim to be heterozygous.

Whether this is the truth or a lie it gives them 4 possible genootypes: KO, LL, KL and OO

Douglas later claims that the Knight is homozygous, so he can't be the Knight.

 

April claims that Baxter is a Knave (always heterozygous) and that Douglass is heterozygous. She also claims that there are only two heterozygotes in the group and since the only genotype that would allow her to be a Knight is KO (given her earlier claim of being heterozygous) these statements can't all be true and so she can't be the Knight.

 

That leaves Baxter and Celeste as candidates for Knighthood.

 

Baxter is harder to rule out (or I can't see the easy way). If Baxter was the Knight, then the liar would be an LO genotype (which April and Douglass cannot be – see above) making Celeste the liar. She claims April is a heterozygote, so that would mean she was actually homozygous. The only remaining phenotypes are Knave (always heterozygous) and Transposer) always homozygous, making April a Transposer (OO) and Douglass a Knave (KL).

Since April is a Transposer, her third statement (made after Celeste the Liar) would be true i.e. there are two heterozygotes and two homozygotes. That would mean Baxter would have to have a homozygous genotype (KK). That causes a contradiction… Douglass the Knave claims to be heterozygous in his first statement, which we know to be true, so his second and fourth statements must be false. But his 4th statement is that the Knight is homozygous, which would be true… so Baxter can't be the Knight.

 

That means that Celeste is the Knight. With all Celeste's statements being true we know that:

  • Douglass is April's father.
  • The liar is LL genotype
  • Baxter is Celeste's son
  • April is heterozygous and therefore can only be KL (Knave)

 

We know April's first statement (that she is heterozygous) is true, so we can deduce the following (useful) information from her remaining claims:

  • There are 2 heterozygotes and 2 homozygotes in the group. Since the liar is an LL genotype, Celeste must be a KO.
  • Douglas is homozygous (actually we already knew this, but it's a handy reminder)

 

That leaves Baxter and Douglass as candidates for the open positions of Transposer and Liar.

 

Douglass' last claim, (that the knight is homozygous) is false (Celeste is KO). He says this after April's false statement, so he cannot be the Trasnposer. Douglass is the liar. Knowing this we can infer from his statements that:

  • He is NOT Baxter's Father
  • He is not related to two others by blood

 

By process of elimination, Baxter is the Transposer. From his statements we know that:

  • Celeste is April's mother

 

 

In summary:

 

April: Knave (KL) – Douglass and Celeste's daughter, Baxter's half sister

Baxter: Transposer (OO) – Celeste's son (but not Doglass' son – his father was an OO, KO or LO), April's half brother

Celeste: Knight (KO) – April's mother, Baxter's mother

Douglass: Liar (LL) – April's father


  Posted by scott on 2013-04-12 10:54:27
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