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There is a game where "a suitcase is packed" and you have to remember all the words or objects that are packed into the suitcase.

Each participant adds a new word to the "chain of words" one after the other.

I wonder whether this information is stored in the "long-term memory" or whether it only remains in the "short-term memory".

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Short answer
The game you describe is a textbook example of one relying on working memory.

Background
The game you describe is an example of one that taxes working memory. Working memory can be conceptualized as (Chai et al., 2018):

a short-term storage component with a capacity limit that is heavily dependent on attention and other central executive processes that make use of stored information or that interact with long-term memory.

Short term memory being (Cowen, 2009):

[F]aculties of the human mind that can hold a limited amount of information in a very accessible state temporarily.

The difference between working memory and short-term memory is confusing, and Cowen (2009) continues by disambiguating working memory and short term memory:

[W]orking memory includes short-term memory and other processing mechanisms that help to make use of short-term memory.

However, Cowen (2009) points to another definition that is:

the attention-related aspects of short-term memory. This, however, is not so much a debate about substance, but rather a slightly confusing discrepancy in the usage of terms.

Interestingly, when you look up the Wiki page on memory span, which describes the digit span test, a textbook example of a working-memory test, you can see that they also use the terms short-term memory and working memory interchangeably. So I reckon the exact differences between the two are indeed not always clear.

Now long-term memory is another matter altogether, and is, according to the current concepts about memory, not needed for your game, other than to associate and give meaning to the items added (remembering the attributes in the suitcase in a foreign language unbeknownst to you would certainly complicate matters!). Long-term memory being

a vast store of knowledge and a record of prior events.

The difference with short-term storage is that it has a longer decay time and a larger capacity. However, the exact differences remain a matter of debate (Cowen, 2009).

Regardless the exact differences, conceptually, the game relies on working memory.

References
- Chai et al., Front Psychol (2018), 27
- Cowen, Prog Brain Res (2008); 169: 323-38.

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Short-term memory capacity is very limited (How much can users hold in their short term memory), and that seems to be a fundamental property of short-term memory (Can working memory actually be improved?). So the longer this game progressed, the less likely it would be that players could keep the information in short-term memory. A few items would be fine. Ten items would be difficult, and items would start to drop out of short-term memory.

Memory experts use techniques like the method of loci to associate new information with an existing structure held in long-term memory. So they would have a significant advantage in this game.

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