**From:** Anton Cox (*A.G.Cox_at_city.ac.uk*)

**Date:** Thu Sep 27 2001 - 07:10:47 PDT

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

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