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 Genetics of Colors
How to foresee the probability to have brindle or fawn puppies in a litter.
 Some definitions:

Phenotype : what we can see, what express itself
Genotype  : what we don't always see !  what is here but don't express itself.

Homozygote: this is told about a dog who carriing only one gene  ( the phenotype and the  genotype are the same ) 
Hétérozygote : this is told about a dog who carriing two différent genes (the phenotype and the genotype are different )


To cut short we are going to call the colors by letters. We write in capital letters the dominante colors and in small letters the  recessive colors.
So: "B" for brindle and" f " for fauwn Each dog carry two genes for the color ( for this subject)). and in a mating each dog give one of his two genes to the puppies..

So, a fauwn dog is "ff" ,the fauwn is recessive so a fauwn dog is homozygote. Because if he was heterozygote ( two fifferent genes) he will be "Bf" and as the brindle is dominante he will be brindle!!
A brindle dog can  be :
eather "BB" Brindle( homozygote)he is carring only brindle and can give only this color to his puppies.
or "Bf" Brindle ( Heterozygote) he is carriing fauwn.  He can give either the brindle gene or the fauwn one to his puppies.

 So we have 3 solutions about the parents.
 "ff" = fauwn
"Bf" = Brindle carriing fauwn
"BB" =  Brindle not carrring fauwn.

Illustration with boards of both parents genes crossings to see the probability in pourcent to have this or that color.
 The mother ( mère) will be wrote in pink and the father ( père) will be wrote in blue.
First mating ,the simpler one, croiss two fauwn (ff) together. Each parent is going to give one of his gene. the male is in blue , the female in pink.
f f
f ff ff
f ff ff
Each of the parents had only fauwn to transmit! so 100% of the pippies will be fauwn!
Second mating, two brindle together. that becomes more difficult!  because we have 3 possible cases! eather both parents are homozygotes ( Case N°1), or both parents are heterozygotes ( Case N°2 ), or one is  homozygote and the other heterozygote ( Case N°3 ).
CASE N°1 : Both parents are  homozygotes (BB) (not carriing fauwn)
B B
B BB BB
B BB BB
Each of the parents had only Brindle to transmit ! So  100% of the puppies will be homozygotes as their parents!They will never give fauwn puppies.

CASE N°2 : Both parents heterozygotes. (Bf) ( carriing fauwn )
B f
B BB Bf
f Bf ff
Watch the 4  central squares of this board. We can  see here the possibility of color repartition. We have 1 square on 4" BB", so 1/4 of the litter will be brindle homozygote (not carriing fauwn). We have 1 square on 4 "ff", so 1/4 of the litterwill be fauwn, and we have 2 squares on 4  "Bf", so half of the litter will be brindle heterozygote(carriing fauwn).

CASE N°3 : One of the parents is  homozygote (BB) and this other is heterozygote (Bf).
B B
B BB BB
f Bf Bf
100% of the puppies will be brindle! but one half will be  Homozygote (not carriing fauwn"BB")and this other one  will be heterozygote ( carriing fauwn "Bf"). Not easy to choose a puppy if you want him to  transmit fauwn when he will be mated.
Third mating, Cross a fauwn dog (ff) with a brindle homozygote one(BB).
B B
f Bf Bf
f Bf Bf
100% of the puppies will be brindle, but all carries fauwn!!! so it is easy to choose your puppy in this litter , all with  produice fauwn puppies if they are well mated.
Fourth and last mating. Cross  fauvn dog (ff) and a brindle heterozygote one (Bf)
f f
B Bf Bf
f ff ff
50% of the puppies will be fauwn and  50%  will be brindle. And the brindle ones will all be heterozygotes (Bf) (carriing fauwn).
So , we  made the turn of the question! of course all of this are possibilities. If you take the last mating, it is posisble to have only brindle or only fauwn puppies by  the struck of luck . If you have 8 puppies out of this mating nothing says that you  will have 4 brindle and 4 fauwn. It is possible that you have 5 brindle and 3 fauwn! 

On this other hand there is some certainly. Two fauwn will always give fauwn puppies, two brindle homozygoteswil always give brindle homozygotes...  
 

And the pied gene?? how does it works????
The pied gene is récessive . The puppy must heritate it from his both parents to express it.As the gene is recessive, we will write it in small letter "c". A pied-fauwn dog will be called "fc fc" , an homozygote pied-brindle will be called "Bc Bc", an caille heterozygote pied-brindle will be called " Bc fc". A dog who has 2 "c" in his gene code will be pied, a dog who has only one "c" in his gene code will be pied carrier.

Color codes recapitulation:

BB = Brindle homozygote (no carriing fauwn and no carriing pied))
Bf = Brindle heterozygote (fauwn carrier)
ff = Fauwn
BcBc = homozygote pied-brindle (not carriing fauwn)
BcFc = heterozygote pied-brindle ( fauwn carrier)
fcfc = Pied- fauwn
Bcf = Brindle heterozygote (fauwn and pied carrier)
BcB = Brindle homozygote (pied carrier only )
fcf = Fauwn carriing pied
Illustration with boards of both parents genes crossings to see the probability in pourcent to have this or that color.
 The mother ( mère) will be wrote in pink and the father ( père) will be wrote in blue.

Two Pied-fauwns mating

fc

fc

fc

fcfc

fcfc

fc

fcfc

fcfc

All the puppies will be pied-fauwn. 

Two  homozygote pied-brindles (not carriing fauwn) mating
Bc Bc
Bc BcBc BcBc
Bc BcBc BcBc
All the puppies will be Pied-brindle and won't carry fauwn.

Mating between a pied-brindle and a pied-fauwn
Case n°1 : cross an Homozygote pied-brindle and a pied fauwn
Bc Bc
fc Bcfc Bcfc
fc Bcfc Bcfc
All the puppies will be pied-brindle carring fauwn ( and pied-fauwn)

Case n°2 : cross an heterozygote pied-brindle and a pied-fauwn
Bc fc
fc Bcfc fcfc
fc Bcfc fcfc

50% of the puppies will be  heterozygote pied-brindle(fauwn carrier) and 50% of the puppies will be pied-fauwn.
Mixte mating ( a pied dog with a solid one)

Mating of a brindle with a pied -brindle.

Case n°1: an non carriing pied homozygote brindle and an homozygote pied-brindle(non carriing fauwn) .
B B
Bc BBc BBc
Bc BBc BBc

All the puppies will be homozygote brindle and they all carry pied gene.

Case n°2 : an heterozygote brindle non carriing pied and an homozygote brindle ( fauwn carrier).
B f
Bc BBc Bcf
Bc BBc Bcf

All the puppies will be brindle carriing pied but only 50 % will carry fauwn.

Case n°3 : an heterozygote brindle non carriing pied and an heterozygote pied-brindle (fauwn carrier).
B f
Bc BBc fBc
fc Bfc ffc

All the puppies will carry pied. 75% of the puppies will be brindle  ( 2/3 will be heterozygotes and 1/3 homozygote). 25% of the puppies will be fauwn (pied carrier) 

Case n°4 : an  homozygote brindle non carring pied  and an heterozygote pied brindle (fauwn carrier)
B B
fc Bfc Bfc
Bc BBc BBc

All the puppies will be brindle and pied carrier but only  50 % of the puppies will be fauwn carriers.

Case n° 6 : a  pied carrier homozygote brindle and an homozygote pied brindle(non carriing fauwn)
Bc B
Bc BcBc BBc
Bc BcBc BBc

50% will be pied- brindle  and 50% of them will be pied carrier brindle.

Case n°6 : a pied carrier homozygote brindle and an heterozygote pied-brindle (fauwn carrier).
Bc B
Bc BcBc BBc
fc Bcfc Bfc

50% of the puppies will be pied-brindle with  half fauwn carrier,, 50% of them will be pied carrier brindle with half fauwn carrier.

Case n°7 : an pied carrier heterozygote brindle (fauwn carrier)  with an homozygote pied-brindle(non carriing fauwn)
Bc f
Bc BcBc Bcf
Bc BcBc Bcf

or
B fc
Bc BBc fcBc
Bc BBc fcBc

50% of the puppies will be pied carrier brindle with half fauwn carrier inside. 50% will be pied-brindle with half fauwn carrier inside.

Case n°8 : a pied carrier heterozygote brindle (fauwn carrier) with an  heterozygote pied-fauwn (fauwn carrier).
Bc f
Bc BcBc Bcf
fc Bcfc fcf

or
B fc
Bc BBc fcBc
fc Bfc fcfc
This mating can give all the colors. 

Mating of a brindle and a pied-fauwn.

Case n°1 : A homozygote brindle (non carriing fauwn and pied)with a pied-fauwn.
B B
fc Bfc Bfc
fc Bfc Bfc

All the puppies will be fauwn and pied carrier brindle .

Case n°2 : A pied carrier homozygote brindle  and a pied-fauwn.
Bc B
fc Bcfc Bfc
fc Bcfc Bfc

50% of the puppies will be pied and fauwn carrier brindle and 50% of them  will be fauwn carrier pied-brindle.

Case n°3 : An heterozygote brindle (fauwn carrier) non carriing pied with a peid-fauwn
B f
fc Bfc ffc
fc Bfc ffc

50% of the puppies will be pied carrier heterozygote brindle (fauwn carrier) and 50% ot them will be pied carrier fauwn.

Case n°4 : pied carrier Bringé heterozygote brindle (fauwn carrier) with a peid-fauwn.
Bc f
fc Bcfc ffc
fc Bcfc ffc

or
B fc
fc Bfc fcfc
fc Bfc fcfc
This mating can give all  the colors.

Mating between a Fauwn and a pied-brindle
Cas n°1 :  a fauwn with an homozygote pied-brindle (non carriing fauwn).
f f
Bc Bcf Bcf
Bc Bcf Bcf

All the puppies will be pied carrier heterozygote brindle (fauwn carrier).

Case n°2 : a fauwn with an heterozygote pied-brindle (fauwn carrier).
f f
Bc Bcf Bcf
fc fcf fcf

50% of the puppies will be  pied carrier heterozygotes brindle (fauwn carrier) and 50% of the puppies will be pied carrier fauwn.

Mating between a fauwn and a peid-fauwn.

Case n°1 : A Fauwn non carriing pied with a pied -fauwn.
f f
fc ffc ffc
fc ffc ffc

All the puppies will be pied carrier fauwn.

Case n°2 : a pied carrier fauwn and a pied-fauwn.
fc f
fc fcfc ffc
fc fcfc ffc

50% of the puppies will be pied-fauwn and  50% of them will be pied carrier fauwn.