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Would a braille watch secreted in one's pocket not suffice for this effect?

Shush!

How about creating a device for this?

Most likely you will already know what the hour is. If you were in a torpor - at work for example - you can get away with it by saying "x minutes" past or "x minutes" to.

So the only thing you need is the minute hand on a watch.

I wonder if you could buy an inexpensive wrist watch, remove the mechanism from it, and attach a shirt button to it on which you would indent a mark.

Then the watch mechanism would be hidden behind your shirt, leaving the button to protrude.

Or could the same principle be applied to a circular lapel pin, with the mechanism hidden behind the lapel?

Vanderbelt, it's from Crocodile Dundee. But he didn't had a watch under his hat, simply looked at his friend's wrist watch to impress the lady.

When somebody asks me if I know the time I simply say 'yes' and walk away. but I'm a butt head.

Know the time? I hardly know what day it is!

Jerome Finley in one of his works, Random acts of kindness, talks about an effect called Godspeed, where time appears to speed up for a participant. Its a bit of a long play and you are not there to see the pay off and in effect you are taking credit for a natural phenomenon before it actually happens, like cloud bursting or even picking which lift will arrive first I guess.