[Inquiry] Re: Musement On Invariants

[Jon Awbrey]

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MOI.  Note 2

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For the sake of this musement, let's define
an "invariant" as a function of variables
whose functional value varies less than
its variables do.

For example, consider what happens when you add the digits
of a two digit number and subtract the resulting sum from
the original number.  A two digit number is a number in
the range 10 to 99, in other words, a number of the
form 10a + b, where a is between 1 and 9, inclusive,
and b is between 0 and 9, inclusive.  So the sum of
the digits is a + b, and subtracting that from the
original number yields (10a + b) - (a + b) = 9a,
in the upshot, a number that is divisible by 9,
that is, 9, 18, 27, 36, 45, 54, 63, 72, or 81.

This is the gist of the mind-reading gimmick at this site:
http://www.niehs.nih.gov/kids/mindread/psychicSparkle.swf

The invariant in question is the function f(a, b) = 9a,
where the ordered pair <a, b> varies through 90 values
while the value 9a varies through only 9 values total.

This accounts for the surprise value of the trick,
since our default expectation is that we can vary
the outcome of the computation to the same extent
that we vary our initial choice, but the property
of the invariant function defeats this assumption.

All the designer of the software had to do was make sure
to cover all of the indicated multiples of nine with the
same symbol that would be pre-destined to show up in the
crystal, for the sake of misdirection randomly assigning
any symbols to the remaining numbers less than a hundred.

TGIF !!!!!

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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