Philip J. Fry inherited two-thirds of his genes from his mother and one-third from his grandmother. How is this possible?

In the following equations, F represents Philip, M his mom, D his dad, and G his grandmother.

- F = 1/2*M + 1/2*D
*Fry gets half his genes from his mom and half from his dad.*
- D = 1/2*G + 1/2*F
*Dad gets half his genes from his mom (Grandma Mildred) and half from his dad (Fry himself).*
- 1/2*D = 1/4*G + 1/4*F
*Divide equation (2) in half.*
- F = 1/2*M + 1/4*G + 1/4*F
*Substitute equation (3) into equation (1).*

Therefore, Fry gets half his genes from his mother, a quarter from Grandma Mildred, and a quarter from himself. This is where it gets tricky:

- F = 1/2*M + 1/4*G + 1/4*F
*Original equation.*
- F = 1/2*M + 1/4*G + 1/4*(1/2*M + 1/4*G + 1/4*F)
*Substitute the original equation into itself.*
- F = 1/2*M + 1/4*G + (1/8*M + 1/16*G + 1/16*F)
*Distribute the 1/4*
- F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F
*Group like terms.*

- F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F
*New equation.*
- F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*{(1/2+1/8)*M + (1/4+1/16)*G + 1/16*F}
*Substitute the new equation into itself.*
- F = (1/2+1/8)*M + (1/4+1/16)*G + {1/16*(1/2+1/8)*M + 1/16*(1/4+1/16)*G + 1/16*1/16*F}
*Distribute the 1/16.*
- F = (1/2+1/8)*M + (1/4+1/16)*G + {(1/32+1/128)*M + (1/64+1/256)*G + 1/256*F}
*Distribute the 1/16 again.*
- F = (1/2+1/8+1/32+1/128)*M + (1/4+1/16+1/64+1/256)*G + 1/256*F
*Group like terms.*

Continue indefinitely to get:

F =

(1/2+1/8+1/32+1/128+1/512+1/2048...)*M +

(1/4+1/16+1/64+1/256+1/1024+1/4096...)*G +

(

*an infinitesimally small fraction*)*F.

The sum (1/2+1/8+1/32+1/128+1/512+1/2048...)

converges to 2/3. The sum (1/4+1/16+1/64+1/256+1/1024+1/4096...) converges to 1/3. Therefore, Fry's genetic makeup is 2/3 from his mother, 1/3 from his grandmother, and an

infinitesimally small

Y chromosome.

QED.