| 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 |
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.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1873.1 | Eleven, and you get one large and one small pizza | GEMGRP::NOYCE | DEC 21064-200DX5 : 138 SPECint @ $36K | Tue May 24 1994 22:08 | 20 |
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.
| |||||
| 1873.2 | RUSURE::EDP | Always mount a scratch monkey. | Tue Jun 13 1995 14:05 | 8 | |
11 is the published solution.
-- edp
Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.
| |||||