
merken
to notice
To perceive, notice, or realize something; to become aware of something.
Ich habe erst zu spät gemerkt, dass ich meinen Schlüssel vergessen habe.
I only noticed too late that I had forgotten my key.
to notice / remember
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to notice
To perceive, notice, or realize something; to become aware of something.
Ich habe erst zu spät gemerkt, dass ich meinen Schlüssel vergessen habe.
I only noticed too late that I had forgotten my key.

to remember
To keep a piece of information in memory; to remember something.
Ich kann mir Telefonnummern einfach nicht merken.
I just can't remember phone numbers.

to let show
To let one's feelings, thoughts, or intentions be recognized through one's behavior or facial expression. Usually used in the negative.
Obwohl er sehr nervös war, ließ er sich nichts anmerken.
Although he was very nervous, he didn't let anything show.
This verb changes its meaning depending on whether it is reflexive or not. 'Etwas merken' (to notice something) means to perceive something. 'Sich etwas merken' (to remember something) means to memorize something or keep it in memory.
Ich merke, dass es kälter wird.
I notice that it's getting colder.
The structure for 'sich etwas einprägen' (to memorize something) is always: `sich (dative) + etwas (accusative) + merken`. The person who remembers something is in the dative, and the information is in the accusative.
Merk dir bitte meine neue Adresse.
Please remember my new address.
In the meaning 'wahrnehmen' (to perceive), 'merken' (to notice) is very often used with a subordinate clause that begins with 'dass' (that). This construction expresses that someone realizes a situation or fact.
Er hat gar nicht gemerkt, dass alle Leute ihn ansahen.
He didn't notice at all that everyone was looking at him.
The expression 'sich etwas anmerken lassen' (to let something show) is a fixed expression. It is usually used in the negative ('sich nichts anmerken lassen' — to not let anything show) to express that someone successfully hides their feelings.
Sie war sehr aufgeregt, aber sie hat sich nichts anmerken lassen.
She was very excited, but she didn't let anything show.
to recognize / realize
Both words can express becoming aware of a mistake or deception. merken initially means that something catches your attention; erkennen emphasizes more strongly that you correctly understand the content or the deception. merken fits simply becoming aware of something, while erkennen fits the insight you gain.

An den widersprüchlichen Zahlen merkte ich, dass etwas im Bericht nicht stimmte.
The contradictory figures made me notice that something was wrong in the report.
An den widersprüchlichen Zahlen erkannte ich den Fehler im Bericht.
The contradictory figures made me realize that there was an error in the report.