I Marvel @ ...

I marvel at

(1) the magical flexibility in the applications of the Pigeonhole Principle (PP).
The PP may be simply stated as..
Let m and n be any two positive integers. If m objects are distributed among n boxes, then one of the boxes must contain at least [m/n] objects.
[x] denotes the least integer not less than x. { E.g. [1.2] = 2, [3.01] = 4, [2.5] = 3, etc. }
Note : [.] is not the standard mathematical notation. It's my limitation for not being able to create the kind of brackets without the "little horizontal" line at the bottom.

A simple illustration :
Suppose 5 (m = 5) pigeons are to be put into 2 (n = 2) compartments. PP says that at least one of the compartments will contain at least 3 pigeons.

*Ahem. Apparently, it's so common sense! Do we really need PP to tell us "at least one of the compartments will contain at least 3 pigeons?"
So PP must be so trivial, so common sense then? Not really.

Try taking on the little challenge to answer the following questions :
Q1. Show that, among any four integers, there are two integers whose difference is divisible by 3.
Q2. Seven darts are thrown at a dartboard which is circular and of radius 20 cm. If all the darts land within the dartboard, show that there are two darts not more than 20 cm apart.
Q3. Show that if a and b are integers, then the decimal representation of a/b either terminates or eventually, a block of digits repeats itself to infinity. For example, 6/5 = 1.2 and 12/7 = 1.7142857142857142857... .

The above questions (and plenty more advanced challenges) can be elegantly solved by applying PP sensibly and logically. Simply magical. No worries, I wouldn't want to bore you with the solutions. However, this is to be taught next year to some selected students (based on the new syllabus). Crazy! Erps.. I'm referring to the idea of teaching something of such level to JC students. To me, the profoundness is comparable to university materials.


(2) the ability of some of my *little pals who can write concisely and precisely (yet not necessarily using bombastic words). Indeed, there are many great writers writing in a style using the simplest words and yet suffers no lost in precision and can evoke the intended feelings and emotions within us, at the same time making us think. Ha.. My way of writing simply pales in comparison and frankly, these *little pals are the ones giving me the initial motivation to maintain a blog. Anyway, they somewhat spur me to continue to improve on.