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

Using a disposable glove, spread the paste on the blackened grout and scrub with a brush or sponge. Leave for about 10 minutes and rinse with warm water. If necessary, repeat the operation until the desired surface is completely whitened.

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.