extra bits to a message, Shannon showed that it is pos-
sible to detect and correct errors that occur during the
transmission of messages due to noise.
Shannon accepted a faculty position at the Mas-
sachusetts Institute of Technology in 1957, but
remained a consultant with Bell Telephones. He contin-
ued his work on Boolean algebra, applying it to the
new field of artificial intelligence. Shannon produced
the first effective chess-playing programs.
Shannon received many honors for his work,
including the National Medal of Science in 1966 and
the Audio Engineering Society Gold Medal in 1985. He
died in Medford, Massachusetts, on February 24, 2001.
Sicherman dice A pair of dice with faces renumbered
1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8 are called Sicherman
dice, named after their discoverer Col. George Sicher-
man of Buffalo, New York. These dice have a remark-
able property: the
PROBABILITY
of throwing any
particular sum (from 2 to 12) with Sicherman dice
matches the odds of throwing that same sum with a
pair of standard dice. Thus Sicherman dice can be used
in dice games without affecting the odds of the game.
The tables below show all possible outcomes of
rolling a pair of standard dice (top) and a pair of
Sicherman dice (bottom), 36 combinations in all for
each pair. All sums appear equally often in each table,
and so indeed all sum probabilities are identical. (Both
tables possess three 10s, for instance. The odds of
throwing a 10 are thus the same for each pair: or
).
The Sicherman dice are unique: there is no other
way to renumber the faces of two cubes with positive
integers yielding two dice with the same sum probabili-
ties as standard dice.
Tetrahedral dice (with each die the shape of a
TETRAHEDRON
) having faces labeled 1, 2, 2, 3 and 1, 3,
3, 5 produce the same sum probabilities as ordinary
tetrahedral dice labeled 1, 2, 3, 4 and 1, 2, 3, 4.
sieve of Eratosthenes (Eratosthenes’ sieve) Although
there is no known formula for generating
PRIME
num-
bers, there is a simple method for “sifting out” the
primes between 1 and any given number N. The proce-
dure is called the sieve of Eratosthenes and is attributed
to 3rd-century scholar E
RATOSTHENES OF
C
YRENE
, a
Greek contemporary of E
UCLID
(ca. 300–260
B
.
C
.
E
.).
The method is performed as follows:
1. List all the positive integers from 2 up to N.
2. Leave the number 2, but cross out every second
number after it. This deletes all the multiples of 2
greater than 2.
3. Leave the next remaining number, 3, but cross out
every third number after it (if not already deleted).
This will delete all the multiples of 3 greater than 3.
4. Leave the next remaining number, 5, and delete all
of its multiples.
5. Continue this process by always going to the next
remaining number and crossing out the multiples of
it that occur further along in the list.
6. The integers not deleted when this process ends are
the prime numbers between 2 and N.
Any number kin the list that is not prime can be fac-
tored, k= a×b, and so will be deleted when consid-
ering the multiples of a, say. Only those numbers in
the list that do not factor, that is, the prime numbers,
will survive.
1
––
12
3
––
36
462 Sicherman dice
Comparing ordinary dice with Sicherman dice