Why x*0 = x

[Alex K]

You mess up your own pattern at the end. Let's define the pattern using algebra:

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x * y = x + ((x - 1) * y) + (y - x)

Plugging in x=5, and y=6:

5 * 6 = 5 + (4 * 6) + 1
5 * 6 = 5 + (6 + 6 + 6 + 6) + 1

Plugging in x=2, and y=4:

2 * 4 = 2 + (1 * 4) + 2
2 * 4 = 2 + 4 + 2

Seems to match your pattern, right?

Let's try x=0, y=1

0 * 1 = 0 + (-1 * 1) + 1
0 * 1 = 0 + -1 + 1
0 * 1 = 0

Tada! Math still works, the universe isn't broken. If you forget the middle part of your pattern, which is essentially "add m to itself n-1 times" then you don't get a correct answer.

By the way, your pattern simplifies to:

x * y = x + ((x - 1) * y) + (y - x)
x * y = x + (xy - y) + y - x
x * y = xy

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So you can prove its the correct pattern using algebra anyway.

Oh my dog! I was going to do this exact construction, and now you done did it. Chapeau