The main DVD commentary track for the Futurama episode Roswell That Ends Well briefly discusses the immensely complicated situation created in Philip J. Fry's DNA when he accidentally becomes his own grandfather, by travelling back in time to 1947 and sleeping with his paternal grandmother, thereby conceiving his father, Yancy Fry.

Here's Fry's family tree.

Fry = Mildred
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  Yancy = Fry's mother, first name unknown, maiden name Gleisner
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       Fry

Figuring out Fry's genetic makeup

Pick any one of Fry's genes and follow it forwards in time. Call this Gene A.

When Fry sleeps with Mildred in 1947, there is a 50% chance that Gene A gets carried on to Yancy, and a 50% chance that Gene A is discarded. That means Yancy's genetic makeup is 50% Fry's genes and 50% Mildred's genes.

Assume that Gene A did make it to Yancy. Now go forward another generation. We know that Fry is made up of 50% Yancy's genes and 50% Fry's mother's genes. But we also know for a fact that Gene A is present in Fry's genetic makeup.

That means, for Gene A, either:

1. Fry inherits Gene A from Yancy and it becomes a causal loop: Yancy inherits it from Fry, Fry inherits it from Yancy, and so on forever. Let's say X% of Yancy's genes do this.
2. Yancy does NOT pass Gene A on to Fry - but Fry receives an identical Gene A from his mother instead anyway. Then (50 - X)% of Yancy's genes do this.

Explain that again?

• 50% of Fry's genes are regular genes, inherited from his mother
• X% of Fry's genes are eternal genes with no origin, inherited from Yancy
• (50 - X)% of Fry's genes are regular genes, inherited from Yancy, who originally inherited them from Mildred.

When Fry sleeps with Mildred, the eternal genes are among those Fry passes on:

• 50% of Yancy's genes are regular genes, inherited from Mildred
• X% of Yancy's genes are eternal genes with no origin, inherited from Fry
• (50 - X)% of Yancy's genes are eternal genes with no origin, inherited from Fry, who originally inherited them from his mother.

When Yancy sleeps with his wife and conceives Fry, the eternal genes are among those Yancy passes on:

• 50% of Fry's genes are regular genes, inherited from his mother
• X% of Fry's genes are eternal genes with no origin, inherited from Yancy
• (50 - X)% of Fry's genes are regular genes, inherited from Yancy, who originally inherited them from Mildred.

etc.

Consequences

• The first major question this raises is: what is X? How many of these genes are eternal? This value can't be more than 50%, as only 50% of one's genome gets passed on to the next generation.

  However, we also know X can't be zero. Proof: Fry is male, but all his direct ancestors are ultimately female (apart from himself). That means he has nowhere to inherit the male Y-chromosome from except himself. So his Y-chromosome makes up part of the genes which are in the causal loop.

  Beyond that it is impossible to say what X is.

• Fry and Mildred have (50 - X)% of their genetic code in common. Depending on what X is (see above), this is either disgustingly incestuous (for X ≅ 5) or entirely inoffensive and legal (for X = 50).

  But remember that the situation with Fry is exactly symmetrical with the situation with Yancy. Yancy is also his own grandfather. And Fry's mother and Yancy also have (50 - X)% of their genetic code in common, which makes their relationship just as bad!

• It seems reasonable to assume that the eternal genes comprise more than just Fry's Y-chromosome. Fry is unique in the entire human species in that he lack the delta (intelligence) brainwave, instead cobbling together other brainwaves as a substitute, which makes him immune to the intelligence-draining powers of the evil brainspawn he fights in the episodes The Day The Earth Stood Stupid and The Why Of Fry. This could be either due to the eternal genes or due to the fact that he may be rather inbred. It seems unlikely that Fry would be the only inbred person in the future so we can probably put this phenomenon down to the eternal genes too. (Or perhaps a combination of the two.)

• It is entirely possible that Fry's brother, Yancy Jr., has some of the eternal genes in his genetic makeup as well.