Sleepless Mathematics

You know those nights where you lay in bed for hours and just can’t fall asleep?  When you watch the clock tick your precious moments of sleep away and know you’re already going to be a zombie the next day because you have to get up in a few hours but your body just won’t get comfortable and you can’t seem to nod off?  This was one of those nights.

I got to bed at the reasonable hour of 10:30.  Usually I’m at the moment or before my head hits the pillow but not this night.  After spending what seemed like forever convincing myself that I was indeed falling asleep I looked up the clock to see how little time I had left to rest.  It was 12:30.

Aside from the fact that I had to wake up in five hours, there was something interesting about 12:30.  I have a habit of making equations about the time of day, and 12:30 was especially special because I could write an equation without even rearranging the numbers: .  I wish I could have taken back that thought, but it was too late; a new math game was in the works.  The question is whether or not I can make an equation which equals the last digit of the time by using the first three in order.  By the time I had thought of the rules of the game it was 12:31.  Here is what I came up with:

Then 12:40 came and I became stuck.  I’ve thought about it a little bit and I think that 12:3_ is the only 10 minute chunk where you can make all the digits using the first ones, but I’m not sure.

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