modular arithmetic

From: Anton Cox (
Date: Thu Sep 27 2001 - 07:10:47 PDT

I now know that the judge is not alone in thinking that it is now
impossible to throw a 5 on a single die. I disagree - here's why.

The relevant text says:

   "From now on, when the dice are thrown, the number that comes up on
    each die must be equal to the number of letters (modulo 5)...".

I know that some believe that this means they must equal 0,1,2,3 or 4.
But that is not the case. Modular arithmetic can be regarded as
working with representatives of equivalence classes. For example, the
numbers 1, 6, 11, 16 etc are all elements of the same class - which is
commonly denoted by the symbol 1. But we could just as easily take
1,2,3,4,5 (or even 1,7,3,24,35) as our representatives of the five
classes, it makes no difference.

So (unless you specify a set of five elements) the only way to
interpret a statement of the form above is as a statement about
equivalence classes, ie the phrase "must be equal to" is the part of
the sentence that is modified by the caveat "modulo 5", not the part
"number of letters". (Because, to reiterate, the latter does not make
sense if we do not specify the set of representatives that we will be
using.) Why shouldnt I say that you cannot roll a zero because I use
representatives 1,2,3,4,5 (another common choice)?

So if a word of 5 letters (or 10, 15, etc) occurs at the start of the
rule, the next rule can use it to justify rolling either 0 or 5, as
the are both equal to 5 modulo 5.

This also agrees with the usual english usage of the phrase above. If
I remove the part "from now on.. ..thrown" and replace "the number
that comes up on each die" by "10" and "the number of letters" by "20"
then I get the sentence

     "10 must be equal to 20 (modulo 5)..."

which seems relatively uncontroversial to me!

Best Wishes,


Rule Date: 2001-09-27 14:10:08 GMT

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