As an aside, pure tortoiseshells (no white) are rare, as their colouring is a genetic slip-up. Many vets recommend allowing them to have at least one litter.
Umm, not quite. There is no genetic slip up - and it's quite simple really.

Brindle torties are far from rare.
Firstly, the gene for white spotting (S) is independent of other genes for colour / pattern. A cat is just as likely to not have white spotting as having it.
Tortoiseshell colouring is sex-linked and only (other than in some very rare cases) occurs with female cats. This is down to the fact that the y chromosome is too small to hold any genetic information for colour - colour genes are only found on the x chromosome.
Hence, a male cat can either only be orange (O) or non-orange (o) i.e. ginger or not, since as an xy, he only has a single x chromosome.
Female cats have two x chromosomes and therefore have two sets of genes pertaining to colour.
For a female cat to be ginger, she must have *two* copies of the O gene, one on each x chromosome i.e. she is OO
If a female cat only has one copy of the orange gene, i.e. she is Oo, then she will be a tortie of some description, with the non-orange part of her colouring being solid or tabby.
And if a female cat has no copy of the orange gene i.e. she is oo, then she will be black or tabby or some variant thereof.
