Tuesday, September 12, 2006

Solution to Saturday’s puzzle:

Define ƒ:

ƒ(n) is the element that has atomic number corresponding to the n-th prime number, expressed in Chinese.

For simplicity of analysis, ƒ can be thought of as a composite function
ƒ(n) = g(h(n))
ƒ(n) = gh(n)

g(n) = the (Chinese) name the element with atomic number n, and
h(n) = the n-th prime number

Example:
Let n = 25,
ƒ(n) = g(h(25))

h(25) = 97 (the 25th prime number is 97)
g(97) = 锫 (Berkelium has atomic number of 97)

ƒ(25) =锫

Estimate the domain of ƒ:

Elements in with atomic numbers greater than 111 have not been named yet, placeholders are used to refer to these elements. It is analogous to calling a carburettor a ‘thingy’ when one does not know the name.

The greatest prime less than or equal to 111 is 109, which is the 29th prime number. As such, the domain of ƒ lies in [1, 29].


To be more rigorous, let’s break ƒ into its constituent functions g and h.
g(n) is the name of the element with atomic number n.
h(n) is the n-th prime number.

g is defined for all values of n between 1 and 111. Since n is a count of protons in an atom, it cannot take negative values. If there is no proton (n = 0), the particle does not qualify as an atom. If the atomic number is 112 or greater, the element is unnamed, and g is not defined.

h is defined for all values of n greater than zero, since there is no such thing as the zero-th prime number. Since the list of prime numbers is sorted by magnitude, h is a monotonously increasing function: h(n+1) > h(n) ∀n ≥ 1.

For g(h(n)) to be defined, h(n) must not be greater than 111.

It is known that h(1) = 2, h(29) = 109 and h(30) = 113. Since h is a monotonously increasing function, it can be deduced that the maximum possible n that will result in ƒ being defined is 29, and all values of n between 1 and 29 will not result in h(n) exceeding 109.

The smallest value of n possible is 1, and the largest is 29. The domain of ƒ is [1, 29].


Further reading:

元素周期表


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