X2, junbuggena.
Barred of either sort (Cuckoo is "smudgy". The smudge is because the birds are fast-feathering and this causes uneven coloration. Barred birds, such as the plymouth barred rock, have a crisper coloration because they're slow feathering) is sexlinked.
Hens have a W and Z sex chromosome pair. Because it's smaller, Z holds fewer genes than the W chromosome. Roosters have two W sex chromosomes. Barring is on the W chromosome. Z does not contain barring. Hens, therefore, can have only one copy of the barring gene.
If the parent rooster is non-barred and the hen is barred:
A male chick inherits the mother's W chromosome, which contains barring, and the father's W chromosome, which has no barring. He has one copy of the barring gene and is barred.
A female chick inherits the mother's Z chromosome, which has no barring, and the father's W chromosome, which also has no barring. The female chick is not barred.
For the same reason, a slow-feathering hen and a fast-feathering rooster will have fast-feathering pullets and slow-feathering cockerel chicks.
A lot of Marans go broody. Hatchery stock generally is less likely to go broody than breeder stock is. Cochins are also very broody.