Biology! help me with inheritance patterns?

Well, one thing we could consider is whether it's a sex-linked trait or autosomal (not sex-linked).

Sex-linked traits are typically on the X chromosome. For most chromosomes people always have two (versions) of each, hence two alleles for each gene, one from each parent. But only women have two X chromosomes. Men have one X and one Y, and the Y typically doesn't have X chromosome alleles on it.

This means that whatever's on the single X chromosome that a man has fixes their trait even if the allele is recessive. That X must be from the man's mother, since his dad's sole contribution to his son's sex chromosomes is the Y, making him male.

Girls get an X chromosome from their dads, and he only has one to give, so whatever's on that chromosome, they get it.

We're given no info about the trait, but since it's shown using blue I'm going to call it 'smurfism'.

In generation I, the man has smurfism whilst the woman doesn't. All three of their daughters have smurfism, but neither of their sons do. That might be a hint that it's sex-linked.

Looking further through the pedigree it seems that the trend continues. Smurfettes have both sons and daughters with smurfism, but male smurfs (papa smurfs if you like) paired with normal females have only smurfettes with no smurfs. (See children 4 to 10 of generation IV and their parents.)

If it is sex-linked, is smurfism dominant or recessive?

If it were sex-linked recessive, smurfettes would have to have a pair of smurf alleles and be unable to pass on a normal X chromosome. So they couldn't have non-smurf sons because the boys' only X chromosomes come from mum. But III, 4 and 13 are counters to that.

Smurfism being sex-linked dominant fits. Note *all* the daughters of a papa smurf are smurfettes (II, 2, 4 & 8 and IV, 4, 6, 8 & 10). The single smurfy X chromosome from dad is enough.