It's not baking soda or vinegar! This inexpensive product is all you need to quickly whiten blackened bathroom tile grout!

The same procedure can be used to whiten the joints between bathroom tiles and floor tiles.

Other remedies

Another effective but more expensive remedy is the toilet limescale remover gel. Just pour the product between the tiles and let it sit for about 20 minutes. The result is excellent, but it is more aggressive and not very respectful of the environment.