Answer: 1/4.

If their parents were two sets of twins they would be as related as siblings.

Say ubercousin C12 has parents M1 and F2 and ubercousin C21 has parents M2 and F1. M1 & F1 are siblings, as are M2 & F2. C12 and C21 share all 4 grandparents and get 1/4 of their genes from each. W.l.o.g. take a grandparent, say the father of M1 & F1. On the portion of genes from him, M1 and F1 overlap on 1/2. It’s easy to see that C12 & C21 will overlap on 1/4 on their respective portions from him. So 1/4*1/4*4 = 1/4. Geneticists should be able to do this…

I think the math doesn’t get too fun without incest.

Sewell Wright developed path models through a pedigree to calculate the relatedness between arbitrary individuals. A quicker way for a whole set of individuals involves a recursive tabular method, where the relatedness between two individuals is a function of the relatedness amongst their 4 parents.

See the section on the numerator relationship matrix in http://www-personal.une.edu.au/~jvanderw/Genetic_properties_of_the_animal_model.pdf

Here is some R code that constructs a pedigree matching that described and relies on a function that will quickly calculate this relationship matrix:

https://gist.github.com/1256404

For any child you think of a single gene-pair on an ordinary chromosome(not on X or Y chromosomes), one copy came from father (say F1,F2), one from mother (M1,M2). Siblings have a 1/2 chance of sharing the identical fathers gene and 1/2 of sharing the mothers gene., so have 1/2 chance of sharing 1 identical gene and 1/4 chance of sharing two identical genes and 1/4 chance of having neither identical genes. For the non-identical genes, the likelihood that they are the same depends on the relative proportions of genes in the population, but is no better than for unrelated individuals.

Of course for identical twins all genes are identical (with occasional mutations).

Now looking down two generations for cousins and ubercousins.

Any child’s father’s copy of a gene is equally likely to have come from either of the paternal grandparents (call these GF1, GF2, GM2, GM2) and the mother’s copy from the maternal grandparents (GF1′, GF2′, GM1′, GM3′).

For ordinary cousins one or other of these is completely different (say the maternal, sharing a common grandfather), so there is a 1/4 chance of having exactly one identical gene.

For ubercousins, they have 1/4 chance of sharing maternal grandparent gene ans 1/4 of sharing paternal grandparent gene, so overall 1/4 chance of sharing exactly one gene (like cousins), but also 1/16 chance of having both genes identical.

As with siblings the non-identical ones could still be the same, just chances like non-related individuals.

In summary:
both identical one identical neither identical
identical twins 1 0 0
siblings 1/4 1/2 1/4
ubercousins 1/16 1/4 5/16
cousins 0 1/4 3/4

That is ubercousins are precisely more related than cousins and less related than siblings as one would guess.

The picture is slightly different for XY chromosome related genes.

The boy’s Y chromosome comes entirely form the father, which in turn comes directly form the grandfather. So, as long as there is no incest, the Y chromosome of both boy cousins and ubercousisn, are no more likely to be the same than any unrelated boys. In contrast boy siblings always have identical Y chromosomes.

Similarly for boy cousins and ubercousins, their X chromosomes come entirely from their respective mothers and hence ultimately maternal grandparents, so (again with no shared grandparents), as unrelated as anyone in the population.

In summary:
Genes on Boys’ Y or X chromosomes
identical not identical
identical twins 1 0
brothers 1 0
ubercousins 0 1
cousins 0 1

For girls the situation is a little more complex, as their X chromosomes do come form both mother and father, and while the sums are a little different it turns out these share in the same way as ordinary genes.

As noted in all cases the non-identical genes may be the same randomly depending on proportions of different genes in population.

In addition, mutations as Barry Star discusses will sightly reduce level of sharing identical genes.

Of course many features come not from individual genes, but gene combinations, others are to do with upbringing, so the non-genetic commonality or differences may be greater or less.

In the interest of scientific open disclosure, you can check out my simulator here:

http://pastie.org/2621140