Nous ________ les cheveux.

Bert, 

this may be the most asked (and answered) question on this site. 

‘Les cheveux’ is the direct object of the reflexive verb in the phrase ‘nous nous sommes brossé les cheveux’, and as the direct object follows the verb, there is no agreement of the past participle. Agreement only occurs when the direct object precedes the conjugated verb. 

Unfortunately this point is not in the lesson you linked, but covered in the lesson on passé composé (and other compound tenses) with avoir.

 Special cases when the past participle agrees (in number & gender) when used with 'avoir' in the compound past in French (Le Passé Composé)

Bert, it's a great question. Laura's answer is very helpful.

I find it easiest if I forget the "usual rules" that apply to pronominal/reflexive verbs, and instead concentrate on the preceding direct object rule. When we're washing ourselves, nous nous sommes lavés, the direct object is ourselves - the second nous - and, since that precedes the participle, the participle agrees.

When we're brushing our hair, nous nous sommes brossé les cheveux, the direct object is our hair, les cheveux. It doesn't precede the participle, so the participle doesn't agree.

PS I think of the preceding direct object rule as "Well, I'm not clairvoyant: I can't make a participle agree if I don't know the gender and number of the direct object", but that may be taking guerilla grammar too far!

Bon courage: there's loads of great stuff here, and you should never be afraid of asking a question.

Thanks for the answers Stevie and Maarten, although unfortunately I'm still confused.  Maarten, I completely understand the rule about agreeing past participle with the DOC only when the DOC comes before the verb.  But that's not the rule I was applying, or even attemping to apply.  Further confusing this is the fact that you linked to a rule about agreeing the DOC with the past participle when using avoir, but here we're not using avoir we're using être.  When using être I understood that the past participle always agrees with the object based on this rule, which lists no exceptions: https://progress.lawlessfrench.com/revision/grammar/agree-past-participle-with-subjects-gender-and-number-with-etre-verbs-in-le-passe-compose-conversational-past. Based on Stevie's answer, however, it sounds like there may be an exception to this rule when we're only using être because it's a reflexive verb?  But that isn't indicated anywhere in the rules on this site, which is I think where the confusion is coming from.

Bert, 

Apologies if my response caused confusion. I kept it brief as there are already more detailed answers to this question under the lesson you linked. 

The grammar point regarding the importance of the direct object preceding or following the conjugated verb is not covered in this lesson on reflexive verbs - I agree that it should be.

However, it is functionally similar to the agreements of past participle with auxiliary avoir in compound tenses, covered in the lesson linked in my previous answer. 

To Stevie’s point on agreement and the need for clairvoyance if it were otherwise, the attached TED clip includes a fantastic, plausible demonstration of an (unprovable) explanation. It is in generally understandable French and is very entertaining.

The specific section on past participle agreement starts at about 6 minutes, but the whole clip is worth viewing.

https://www.youtube.com/watch?v=5YO7Vg1ByA8

Thanks Maarten, I see all of the other questions about this now - for some reason I wasn’t seeing that it had been asked so many times in the past. Apologies for asking the same question again! I also see that it accepted my answer as “almost” because there’s no lesson on this, but I’d actually argue that there is a lesson on it already (the one I asked this question on) but the lesson is incomplete. It offers an overly simplified rule stating that the past participle always matches when that’s not the case at all. Seems like a relatively easy fix that would clear up a lot of the confusion and put an end to all of these repeat questions.

Bert - it is easy to miss the previous answers. It was one of my major complaints when I started using the program - I  regularly commented that frequent questions should have their answers incorporated into the relevant lessons.