Cube is a horror movie directed by
Vincenzo Natali and released in 1997. The film was inspired by an episode of
the popular television series of the end fifties and the sixties, The
twilight zone. The episode in question, issued on December 15, 1961, was
titled Five
characters in search of an exit, a title which in turn was inspired by
Pirandello’s play Six Characters in Search of an Author.
This is the summary of Cube’s plot:
Six persons find themselves inexplicably
in an unfamiliar place, consisting of cubic spaces connected together. As they explore,
they discover that there are 17,576 spaces, which together form a larger cube,
26 small cubes per side; that each space is numbered with three three-digit
numbers; and that some of the cubes (those where at least one of the numbers is
prime or a power of a prime) contain deadly traps, while the cubes marked only with
composite numbers (the product of different primes) are safe. Before being
transferred to the cube (we never learn how) the six characters were engaged
in different activities: a policeman, a crook specialized in escapes, a doctor,
a math student, an autistic genius and the architect of the cube. The autistic
boy has the amazing ability to decompose numbers into their prime factors,
which helps them make sure that cells are safe. The math student says that breaking
a number into its prime factors is very difficult. In the end, only the
autistic boy gets out of the cube alive.
Sometimes,
in my classes, I posed my students the following problem:
What is the scientific mistake in Cube?
The mistake
is obvious. In fact, it is indicated in bold italics in the summary of the
film.- Before entering each of the cells, the
characters must discover if any of the three three-digit numbers in the
entrance is a prime number or the power of a prime. Is this so difficult,
as to require the presence of an autistic genius?
- Obviously the writer of Cube’s script knows something, but it is
all wrong. He has no doubt heard about the RSA encryption algorithm,
widely used for Internet security, which is based on the fact that it is
much harder to break down a composite number into its prime factors, than
multiplying the latter to get the first. This is true, for example, for
numbers with 128 figures or more, those used in conventional cryptographic
certificates, but it is not true for three-digit numbers, such as those in
Cube.
- According to the rules of Cube, the three-digit numbers appearing
in the cells are between 001 and 998. Therefore, there are 998 possible
numbers. How many of them are primes or prime powers? 168 are primes; 25
more are powers of primes. The other 805 are composite. What is the
probability that a cell is safe? The cube of the probability that a
three-digit number will be a composite, i.e. (805/998)3 =
52.5%. Not bad: there is almost the same number of safe cells as cells
with traps.
- To find whether a three-digit number is a prime,
one must see that is not divisible by one of the following eleven numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 (the prime numbers less than the
square root of 999). Many of the composite numbers (over one half) can be detected
at first sight; for others one needs to make a few simple mental
calculations, quite easy for a math student (and for many who are not).
- The other dangerous numbers, the 25 prime
powers, are the following:
- Eight powers of 2: 4, 8, 16, 32, 64, 128, 256
and 512.
- Five powers of 3: 9, 27, 81, 243 and 729.
- Three powers of 5: 25, 125 and 625.
- Two powers of 7: 49 and 343.
- Finally, the square of 11, 13, 17,
19, 23, 29 and 31: 121, 169, 289, 361, 529, 841 y 961.
Any math student knows by heart most of these
numbers. The others are very easy to calculate mentally.
In
conclusion, the math student in Cube
should have been able to solve by herself the problem of whether the cells were
safe or unsafe, without the need to introduce the autistic genius. This is the
scientific mistake in the film.
Thematic Thread on Mathematics: Previous Next
Manuel Alfonseca
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