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Large Problem From Below

The context: One of my secondary math student teachers (M.O.) is interning (in both mathematics ans Spanish) at a school in Peru. He recently sent me the following e-mail...



I am observing end of term exams, and at random I wandered into what turned out to be a math exam. I got a copy of the test, and almost entirely did not understand what they were asking for (they're not real big on being crystal clear with definitions, precise wording etc. - another revolution I'm planning for my own class here). But, one problem confused and beguiled me enough to keep working at it.

I'll spare you the Spanish - basically asks, "Calculate the sum of the digits that result from: Sqrt[444...444 - 888...888] (where there are 2000 instances of the digit 4 and 1000 instances of the digit 8).

I worked out a pattern for interior difference, which I found to be unexpected, but believable, but taking the square root to that pattern I found to be especially contrary to my "intuition" regarding square roots. I'm still thinking about why this works.


Your Task: Explore this problem...try to make sense of it. Do your conclusions fit with M.O.'s "brief" observations?

 


Hint: Does it help to focus on patterns related to:
44 - 8
4444 - 88
444444 - 888 etc.
And what is effect of the square root?

Or is there something special about the original problem, separate from the patterns involved?

 


Solution Commentary: In a subsequent e-mail, M.O. added: "As a final answer, the question is to find the sum of the digits that result from the above calculations. Really knocked my socks off, for sure.

I would enjoy publishing your solution and observations...if you will share them....?