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A Mathematical Adventure, or, How I Spent an Afternoon Proving Nothing.

Over the course of a few boring classes last year, I discovered that the product of two numbers plus the square of their difference divided by four is equal to the square of their average:

(a * b) + (a - b) ** 2 / 4 = ((a + b)/2) ** 2

At the time, I thought that it was a pretty neat equation. I couldn't find a use for it, but still was happy about it. I showed my math teacher. She thought it was ... interesting ..., but couldn't say that it had a purpose.

So, today, I was thinking up fun things to do, and I realised that it had been a long time since I played with numbers. I was thinking back to my previous discovery. I thought, "Self, what would I have to add to the product of _three_ numbers to get the square of their average? Wait, make that their cube!"

(a * b * c) + B = ((a + b + c) / 3) ** 3

So, I played around a bit, found out how much of my basic algebra skills had disappeared. It was fun; the feel of the fountain pen in my hand as it left a delicate trail of green ink on beautiful white paper was exhilerating, refreshing. After a while, I found that the bridging factor was the cube of their sum over twenty-seven, less their product:

B = ((a + b + c) ** 3) / 27 - (a * b * c)

I didn't think real hard about that. I went on to four digits, and eventually found that for n digits, the bridging factor was:

B = sum ** n / n ** n - product

Plugging that back into the equation, I found:

product + sum ** n / n ** n - product = (sum / n) ** n
(Remember, average is (sum / n)

I realised that I had goofed; that I had an equality that was basically "x + y - x = y". I felt pretty dumb. I decided to conduct an investigation. I looked back at the equation for two numbers:

(a * b) + (a - b) ** 2 / 4 = ((a + b)/2) ** 2

The bridging factor there concealed the fact that I was simply subtracting the sum; if I had bothered to expand it, I would have noticed:

(a - b) ** 2 / 4 = (a + b) ** 2 / 4 - (a * b)

Silly me! It was good entertainment and stretched my brain, even if the only tangible evidence of my endeavour is a large pile of scrap paper on the edge of my desk.

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