This is so funny! Can I steal that for my profile? 
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On my profile, I ask people to tell me three very random facts about themselves. I’ve gotten such funny and abstruse answers on that! 
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Go ahead! 
Personally, I like more personal questions rather than ones about the “X of your country”. I understand why people like that too… but I go for the personal.
- What did you eat for breakfast today?
- What form of transportation do you use most, and why? (Car, feet, bus etc)
- What time do you go to sleep at night, and when do you wake up?
I have used these on my profile and had some really interesting replies.
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I didn’t know I have a favorite mug before I saw this!
I tend to get bit tired with always writing the same “hey I am…” so thinking of few things I write on cards:
Few lines of song lyrics or part of a poem
Trying to make the text rhyme or write it as a haiku (the plot twist is that I am not good with rhyming)
Write few first lines of a book. This works especially well if the card shows illustration from a book and I can write down few lines from that one.
The favorite day of the year.
What happened on that day some years ago (I keep a day journal)
Where and how I got the card I am sending (it is little difficult to find cards here so sometimes it is a full adventure for me)
Draw stick figures
These are the few I have on my mind now, but I am continually seeking new creative ways to absolutely avoid presenting myself on each card sent and hopefully make the card “bit different”
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How do you ring in the New Year?
What would be your dream holiday?
What do you see when you look out your window?
Is there something that made you smile today?
What did/do you want to be when you grow up?
Does your name have a meaning?
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The other day I did the 5 senses exercise and wrote about it on the back of the postcard.
Write:
Five things you see around you
Four things you can touch around you.
Three things you can hear.
Two things you can smell.
One thing you can taste.
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Your first love 
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I don’t know if this really applies to a writing prompt, but I love to hear about someone’s passion(s) in life!
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Volunteering could be a topic, where people tell about former or current voluntary work, or voluntary work they would like to do.
Wildlife could be another topic.
- (Common) wild animals in the country of the sender.
- Wild animals the sender has seen.
- Wild animals the sender dreams of seeing.
- Or something else related to wildlife.
And meals. For instance: For you, what is a typical…
- …breakfast?
- …lunch?
- …dinner?
- …evening meal?
- What do you eat? Where? With whom?
- What’s your favourite meal of the day?
Or describe the place you’re writing the postcard, using all your senses. And/or describe the view through a nearby window.
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I think one idea I saw on someone’s profile ages ago was to write about some kind of personal habit / ritual that the sender enjoys. So particularly in this time of the pandemic where many of us cannot go out very much, see many people or otherwise live how we used to, I think it could be nice to focus on the positive aspects of daily life, such as:
- What is the one thing / activity that always improves your mood?
- What thing / activity can you not imagine your day about?
- What thing / activity do you now enjoy that you didn’t think you would like?
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I also have a prompt question about “your favorite mug”! [Virtual high five!] I almost included one about neighbors as well, because I currently have atrocious neighbors. 
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I love the wildlife prompt! I should include it in my profile; I enjoy hearing and telling people about local wildlife, because the stories can be really amusing. Because I’m not currently seeing any people during quarantine, observing animal behavior has become a favorite source of entertainment for me!
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- What is the funniest thing you have ever heard or seen?
- What is your favorite childhood memory?
- What are five things on your bucket list?
- What are five cities you want to visit?
- What is the strangest thing you have ever seen or experienced?
- What is an urban legend that is common in your area - like Bigfoot or locally here we have the Lady of the Lake.
- What are three stupid things you are scared of? (Like I am terrified of June Bugs; it’s a whole story.)
- Around holidays I like to know if people have any holiday traditions. Like each year my dad takes the family out to eat on Christmas Eve at the same Italian restaurant and then on Christmas Day after we open presents the kids have an EPIC NERF BATTLE on his ranch and I make up a scavenger hunt where they have to do a funny task at each stop and they get little prizes as they go along. I love to hear about people’s traditions.
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Here’s the list from Postcrossing’s Help page.
https://www.postcrossing.com/help/what-should-i-write-on-a-postcard
- Express what daily life is like where you’re sending the card from by describing what you did today, your routine, etc.
- Write 5 curious facts about the place where the card is from.
- Give local traveling tips from your area! What are the must-sees around you?
- Include your favorite quote, perhaps in its original language (with translation!)
- What was the last thing you cooked or ate? Include a recipe!
- What do you have in common with the recipient of your card?
- Recount a childhood memory, or something you’ve learned from your grandparents.
- What makes you happy? If it makes you smile, there’s a good chance others will like to hear about it.
- Share a local idiom or saying, in its original language, with translation of course!
- Did you ever travel to the place where your card is going? Recall your best memory of that place.
- What’s the weather like as you are writing your postcard? Draw the weather forecast in detail, and don’t forget to include the temperature!
- Share an interesting fact that you’ve learned recently, and which most people are not aware of.
- Got an unusual hobby or collection? Do tell!
- Did anything important happen in your country lately? Share an interesting news tidbit!
- Write about your favorite book, movie or band.
- Draw a picture of your surroundings, or your favorite object in that room.
- Write your favorite poem or…
- …make up your own poem! Acrostic poems or haikus are short and enjoyable to write.
- What are the local festivals or traditions from your area?
- If you’re a student, what are you studying? What was the last thing you learnt in school?
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Besides some that were already mentioned, I have these on my profile:
- What part of your country’s history do you wish more people knew about?
- What is the strangest place you’ve ever slept?
- If you consider yourself artistic, you could draw a sketch of what you think I look like [this only works because I have a non-descriptive profile picture, but I suppose it could be adapted to a more generalized version]
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Are you sure, postcard is suitable for those themes? I mean- the space on the postcard. And must I, in my 60, really remember the last thing I learned in school?
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My favorite questions are:
-What is your favorite fruit or breakfast?
-What’s the last song you’ve heard?
-What’s the last good book you’ve read?
And this one I ask to all new acquaintances when I can:
What two superpowers would you choose and why?
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I would believe it’s possible to write to write about all those topics on a postcard, given that the handwriting is small enough. I know that some find it difficult to write more than 3-4 sentences on a postcard because their handwriting is so large.
Since the suggestions mentioned by @brittanyjewell are all mentioned by the Postcrossing Team in the Help section, I guess they’ve thought it through and only suggested things they know is possible to write about on a postcard. But as I already mentioned, the handwriting is a factor, and most topics can be challenging to write about if one struggles writing small enough.
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