Yes. If I see words, on signs or wherever, I will sometimes mix the letters around to see what other words/interesting-seeming things they make. When I see license plates I'll see if the numbers 'go into' themselves evenly, for instance 954 would because 9 goes into 54 6 times.. Same thing with phone numbers, addresses.. Finding one that works 'evenly' is fun, especially for phone numbers or other long strings of digits.
Also similar things with tiles, like on sidewalks or wherever, see if something can 'zig zag regularly' while only hitting a certain color, or a fun one is to imagine something 'bouncing' along tiles until it repeats, I imagine one tile to be 'active' and it moves at a 45 degree angle (like the bishop in chess) and when it hits an edge it reflects how you'd expect.. at some point it repeats.. sometimes the cycle is short, sometimes it's long.
Also, watching the water drip down while in the shower.. some of the droplets combine, and stream down for a ways, then suddenly they just stop, or essentially disappear. Some don't move for a long time. I sometimes enjoy trying to follow a particular 'glob' of water as it flies out of the shower head.. it's mostly possible.
Really I don't think my mind 'can' be idle, I always have to be thinking of something, even if it's something I've thought of countless times before. Sometimes this is annoying but if I just 'let myself go' it doesn't actually matter. The thing is, this sort of activity, while fun, doesn't actually make me any money, and I have this thing about not 'wasting time' so I typically only do these things when I'm waiting for a bus, showering, although yeah, sometimes I do just drift away.. but it is fun, that's not a waste
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And it just sort of 'happens'
I get the impression that the number of people who do this sort of thing are in the vast, vast, vast minority. What the 'average' person does with all their 'idle thinking time' will forever be a mystery to me. I fear that they don't think about anything at all.. but I don't fear it too much, or else I'd sink into their void
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But perhaps I am entirely wrong and everyone does the same sort of stuff. I've never seen any evidence of this, though.
But yeah, it is fun, isn't it?
I never stop thinking unless I'm in deep sleep, I'm not sure if this is an aspie trait or not. When I'm idle and not doing some pointless mind game, which is the majority of the time I'm idle I'm probably thinking about a special interest or something that for whatever reason is on my mind, or mentally creating/planning something, and sometimes I just get lost in a random imagining. I guess I often think the same stuff quite a bit, but I probably think of at least somewhat more original stuff quite a bit too.
I do the thing with the shower drops, only with rain instead. I don't really think I do what you described with words and numbers a lot, but I believe I do it sometimes.