
der Kopf
head
The uppermost part of the human or animal body that contains the brain, eyes, ears, nose, and mouth.
Seit heute Morgen tut mir der Kopf weh.
My head has been hurting since this morning.
head
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head
The uppermost part of the human or animal body that contains the brain, eyes, ears, nose, and mouth.
Seit heute Morgen tut mir der Kopf weh.
My head has been hurting since this morning.

mind / brain
A person's mind, thinking ability, or intelligence.
Sie ist ein wirklich kluger Kopf und löst jedes Rätsel.
She is a really sharp mind and solves every puzzle.

per head
For each individual person; per person.
Der Eintritt für das Museum kostet zehn Euro pro Kopf.
Admission to the museum costs ten euros per head.

to rack one's brain
To think very intensely and with effort about a problem.
Ich zerbreche mir schon seit Tagen den Kopf darüber, wie ich das finanzieren soll.
I've been racking my brain for days about how I'm supposed to finance it.

to have something in mind
To be thinking about something specific, to have remembered something, or to be pursuing a plan.
Ich habe da schon eine Idee im Kopf, wie wir die Party organisieren könnten.
I already have an idea in mind for how we could organize the party.

head
The frontmost, uppermost, or thickened part of an object.
Du musst den Nagel genau auf den Kopf treffen.
You have to hit the nail exactly on the head.

to lose one's head
To panic in a difficult situation and act without thinking.
Bewahre Ruhe und verliere jetzt nicht den Kopf!
Stay calm and don't lose your head now!

from head to toe
Completely; concerning the whole body.
Er war von Kopf bis Fuß mit Schlamm bedeckt.
He was covered in mud from head to toe.

neck and neck
In a competition or race, being exactly tied or at the same level.
Die beiden Kandidaten liegen in den Umfragen Kopf an Kopf.
The two candidates are neck and neck in the polls.

to give someone a dressing-down
To reprimand someone sharply or criticize them severely.
Nachdem er den wichtigen Termin vergessen hatte, hat ihm sein Chef gehörig den Kopf gewaschen.
After he forgot the important appointment, his boss gave him a proper dressing-down.
In many idioms, 'Kopf' (head) does not refer to the body part, but metaphorically to the mind, thoughts, or intelligence. If you don't understand an expression with 'Kopf', consider whether it refers to thinking.
Sie ist der kreative Kopf des Teams.
She is the creative mind of the team.
The word 'Kopf' (head) is part of many useful compound nouns. Examples are 'Kopfschmerzen' (headache), 'Kopfsalat' (lettuce), 'Kopftuch' (headscarf), or 'Kopfhörer' (headphones).
Ich höre gerne mit Kopfhörern Musik.
I like listening to music with headphones.
In connection with the body part, 'Kopf' (head) is often combined with verbs for gestures or sensations. Typical examples are 'den Kopf schütteln' (to shake one's head / say no), 'mit dem Kopf nicken' (to nod one's head / say yes), or 'Kopfschmerzen haben' (to have a headache).
Er schüttelte nur den Kopf und sagte nichts.
He only shook his head and said nothing.
When 'Kopf' (mind/brain) describes a person, it is often combined with an adjective that names a mental quality. Common combinations are 'ein kluger Kopf' (a clever mind), 'ein heller Kopf' (a bright mind), or 'ein sturer Kopf' (a stubborn person).
Für dieses Problem brauchen wir einen klugen Kopf.
For this problem, we need a clever mind.
The expression 'pro Kopf' (per head/per person) is a fixed phrase from statistics and economic language. It always means 'pro Person' (per person) and is used for average calculations.
Das Pro-Kopf-Einkommen ist in diesem Land sehr hoch.
The per capita income is very high in this country.
The idiom 'sich den Kopf zerbrechen' (to rack one's brain) is reflexive. The person who is thinking uses the reflexive pronoun in the dative ('ich zerbreche mir', 'du zerbrichst dir').
Ich zerbreche mir den Kopf über ein Geschenk für sie.
I'm racking my brain over a gift for her.
The idiom 'den Nagel auf den Kopf treffen' (to hit the nail on the head) uses the meaning 'upper part of an object'. Figuratively, it means that someone says or recognizes something exactly right.
Mit deiner Analyse hast du den Nagel auf den Kopf getroffen.
With your analysis, you hit the nail on the head.
The expression 'Kopf an Kopf' (neck and neck) is used almost only in the context of competitions (sports, politics). It often appears with verbs like 'liegen' (to be placed/stand) or 'sein' (to be), or forms the compound 'Kopf-an-Kopf-Rennen' (neck-and-neck race).
Die beiden Läufer lagen bis kurz vor dem Ziel Kopf an Kopf.
The two runners were neck and neck until shortly before the finish.
In the idiom 'jemandem den Kopf waschen' (to give someone a dressing-down), the person being criticized is always in the dative. This is a fixed grammatical structure.
Der Trainer hat der Mannschaft nach dem schlechten Spiel den Kopf gewaschen.
The coach gave the team a dressing-down after the bad game.
The expression 'von Kopf bis Fuß' (from head to toe) is a fixed phrase and means 'komplett' (completely) or 'am ganzen Körper' (over the whole body). It is often used to describe clothing or a condition.
Sie war von Kopf bis Fuß in Schwarz gekleidet.
She was dressed in black from head to toe.
mind / reason
Kopf and Verstand can both refer to a person's mental ability. Kopf is a more colloquial and figurative term for their intelligence. Verstand directly refers to the ability to think logically, understand, and make judgments.

Mit ihrem guten Kopf löste Nina das schwierige Problem schnell.
With her sharp mind, Nina quickly solved the difficult problem.
Mit ihrem klaren Verstand löste Nina das schwierige Problem schnell.
With her clear mind, Nina quickly solved the difficult problem.