MATH STINGER #8

    by Dr. Joel C. Fowler
    Assistant Professor of Mathematics

    
     The puzzle for this issue is rather unusual and involves
objects that describe themselves in some sense.  For example, the
statement:
               "This sentence has five words."
describes itself.  The puzzle for this issue involves sequences
of numbers that are self-referential.  You are to find a sequence
of 8 numbers: n0, n1, n2, n3, n4, n5, n6, n7 with the property
that:
     n0 = the number of times 0 appears in the sequence,
     n1 = the number of times 1 appears in the sequence,
     n2 = the number of times 2 appears in the sequence,
and so on.  
     For example the sequence:
     n0=2, n1=1, n2=2, n3=1, n4=0, n5=2, n6=0, n7=1
would be incorrect because in that sequence (2, 1, 2, 1, 0, 2, 0,
1) there are 3 ones (rather than 1), there are 3 twos (rather
than 2), there are 0 threes (rather than 1), there are 0 fives
(rather than 2), and there are 0 sevens rather than 1).  The only
digits that are correct are n0 (since there are 2 zeros), n4
(since there are 0 fours), and n6 (since there are 0 sixes).  You
must do more than simply alter the incorrect digits because
changing them affects the correctness of the others.
     Once you have found a correct eight digit sequence there are
many other related questions concerning sequences of this type. 
The following are open ended problems for investigation for your
spare moments during spring break.  Find out as much as you can
or wish for each.  What other lengths are possible for sequences
of this type?  Is there a systematic way of producing sequences
of this type?  Is there more than one sequence possible for a
given length?
     

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