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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1873.0. "Crux Mathematicorum 1946" by RUSURE::EDP (Always mount a scratch monkey.) Tue May 24 1994 15:41

    Proposed by N. Kildonan, Winnipeg, Manitoba.
    
    In a television commercial some months ago, a pizza restaurant
    announced a special sale on two pizzas, in which each pizza could
    independently contain up to five of the toppings the restaurant had
    available (no topping at all is also an option).  In the commercial, a
    small boy declared that there were a total of 1048576 different
    possibilities for the two pizzas one could order.  How many toppings
    are available at the restaurant?
T.RTitleUserPersonal
Name
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1873.1Eleven, and you get one large and one small pizzaGEMGRP::NOYCEDEC 21064-200DX5 : 138 SPECint @ $36KWed May 25 1994 02:0820
    If each pizza can be built N ways, and there's no way to tell them
    apart except for the toppings, then there are N*(N+1) combinations.
    There's no integer N for which N*(N+1)=1048576=2^20.  Thus, there
    must be some difference between the pizzas (perhaps one's large and
    one's small), and there are N^2 combinations.  Each pizza can be built
    1024=2^10 ways.

    I puzzled for a while about how to find sums of quotients of
    factorials that would add up to 2^10.  Finally I came at it this way:
    If there are only 5 toppings to choose from, you can build 32=2^5
    different pizzas (each topping is independently included or not),
    and never worry about having too many toppings on a pizza.  But
    we need many more different pizzas than this.  If there are 11 toppings
    to choose among, you get 2048=2^11 different pizzas -- but some of
    them have more than 5 toppings.  But every pizza with 6 or more
    toppings has a "complement" pizza that has the other 5 or fewer
    toppings, and vice versa, so exactly half the pizzas have too many
    toppings.  Thus the number of legal pizzas is 1024=2^10, and two
    such pizzas can be built in 1048576 ways.
    
1873.2RUSURE::EDPAlways mount a scratch monkey.Tue Jun 13 1995 18:058
    11 is the published solution.
    
    
    				-- edp
    
    
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