Tuesday, October 27, 2009

On the Materialism of Society

When a person is too concerned with material wealth and luxury, we may condemn that person as being materialistic. If we were to take the same argument and apply it not to persons, but to societies, to national entities, then we will arrive at a single conclusion- that Singapore is exceedingly materialistic.

How so? It has been said that the key prerogative of Singapore's leadership is economic growth. And indeed, Singapore has experienced much success towards that goal. But to frame economic growth as the most important of goals, to focus the most energies on enlarging and strengthening the economy, and to then treat the outcomes of such growth as symbols of power, of modernity, of first-world-ness - is there not something missing?

It is sadly laughable for us to not develop the other tenets of society, to hastily ignore that which has no economic benefits, and to then exuberantly claim that "we have arrived!" It does have a hollow ring to it.

It is not to say that the path adopted is wrong, particularly if one does in fact agree with the tenets of materialism. For what could be wrong with more fast cars, more luxury bags, and more cash?

More Tests?

If the aim of assessment is to assess the true ability of a student, then from a statistical point of view, it is better to have more small tests rather than a single big exam.

First, assume that we are using some consistent estimator of a student's ability. The natural conclusion of this assumption is that the more tests we conduct, the more likely it is that the estimate will be close to the true value.

The argument is also intuitive; with more elements of assessment, the impact of a "freak event" that affects student performance is greatly reduced. The result is hence more likely to reflect the true performance of the student, and is fairer due to the reduced role of luck.

Sunday, October 25, 2009

On Consumerism

Richness is not found in excess, nor is poverty found in austerity.

It ought to be obvious that equating material possessions with happiness is, to say the least, unsustainable. It is even more foolish to believe that having more results in a greater happiness; that serves only to move us to the precipice of destruction. A Malthusian accusation perhaps, but one that seems increasingly true as time passes.

I suggest that we look upon our lives with a critical eye, and to discover those things which are truly essential to our happiness. All other things are but luxuries, perhaps enjoyable, but should never be elevated to that of a daily necessity.

Saturday, October 24, 2009

Proofs of Arithmetic Progression


Find as many proofs of the arithmetic progression; i.e. prove 1 + 2 + 3 + 4 + ... + n = 0.5 * n * (n + 1)

I have thought of four proofs, as detailed below.

Proof 1: Proof by Pairing

We pair the first and last elements, then the second first and second last elements, and so on. Each pair will have the same sum, since the earlier element gains 1 while the later element loses 1. Now, divide any pair by half. If there are even number of elements, all elements will be paired, and since each pair is the same, the average is found by diving any pair by half. Now, if there are odd number of elements, the remaining element, lying in the center of the sequence, will have the same value as the half of any pair, and this also yields the average.

Having the average, the sum can be found by simply multiplying by the number of elements, which is n. Hence,

Sum = 0.5 * (first element + last element) * (number of elements)
==> Sum = 0.5 * (n + 1) * (n)

Proof 2: Proof by Mathematical Induction

Suppose that the sum for a sequence (1 + 2 + ... + n) is equal to 0.5 * n * (n + 1) for some value of n = k.

Then, for the sum of a sequence (1 + 2 + ... + (k +1)), we have:
Sum = (1 + 2 + ... + k) + (k + 1)
==> Sum = 0.5 * k * (k + 1) + (k + 1)
==> Sum = 0.5 * (k + 2) * (k + 1)
Hence, if the identity is valid for n = k, it is also valid for n = k+1.

Now, consider n = 1.
Sum = 0.5 * (1) * (1+1) = 1.
Hence, it is true for n=1 ==> true for all n.

Proof 3: Proof by Square

Consider the following image.
A (n+1) by (n+1) square can be broken up into three pieces; n + 1 unit squares, and two arithmetic progressions of (1 + 2 + ... + n). Using this relation,

(n + 1) ^ 2 = (n + 1) + 2 * Sum
==> Sum = 0.5 * ((n + 1) * (n + 1) - (n + 1))
==> Sum = 0.5 * (n) * (n + 1)

Proof 4: Proof by Triangle

Consider the following image.
A triangle of base (n+1) and height (n+1) can be obtained from an arithmetic sequence of of (1 + 2 + ... + n) by adding n + 1 triangles of 0.5 unit area. Using this idea,

Sum = o.5 * (n + 1) * (n + 1) - 0.5 * (n + 1)
==> Sum = 0.5 * (n) * (n + 1)

I hope that you have found this exercise to be entertaining.

Tuesday, October 20, 2009

Coin Problems

You have pennies (1c), nickels (5c), dimes(10c), quarters (25c), and half-dollars (50). Solve the following problems:

a) Choose a combination of exactly 100 coins such that the total sum is exactly $5. There must be at least one of each coin.

b) Choose a combination of exactly 100 coins such that the total sum is exactly $5. There must be at least two of each coin.

c) Choose a combination of exactly 100 coins such that the total sum is exactly $5. There must be at least three of each coin.

d) Choose a combination of exactly 100 coins such that the total sum is exactly $10. There must be at least one of each coin.

Note: For some of the problems, there are multiple possible solutions. It is possible to arrive at a solution by trial and error. However, the ideal way of solving this exercise is to discover a method to solve all the above problems in a less tedious manner than trial and error. =)

Wednesday, October 14, 2009

Rocket Science

A random thought struck me today. In the scenario that a villain is armed with a rocket launcher, and is aiming at you, would it be better to charge forward, or move backward? Assume that no lateral movement is possible, and that the distance is sufficiently close such that the rocket will hit you in seconds at most.

My insight is that moving forward actually reduces the amount of damage you take (if you ignore the salient point that you are likely to die anyway). Note that a rocket is by definition powered. Hence, when in flight, it actually accelerates, which means that if you give it more distance, the rocket gains more speed! The rocket has now a higher kinetic energy on collision. Conversely, by moving forward, the rocket will hit you at a slightly slower speed.

This analysis is, however, not valid for non-powered devices, such as bullets. Bullets will only decelerate, hence it would be wiser to run away in such a case.

This post was inspired by some weapons in comics which gain speed with distance, which on further thought did not seem that improbable after all.

Tuesday, October 13, 2009

On Envy, Revisited

If we were to ignore the cardinal mistake of blaming our woes on others, then perhaps we will view Envy in another light. Contrary to some lines of thought, Envy is not so much an indicator of insufficiency but rather a realization of injustice.

When one has Envy, one does not think "I want X (which Y possesses)". On the contrary, one believes in one or both of two things; in Envy's benign form, one believes that "I deserve X, as I am at least as deserving of X as Y"; in Envy's malign form, one simply believes that "Y does not deserve X". Hence, Envy emerges from the perception of injustice.

Sunday, October 11, 2009

On Envy

Envy is thought to be undesirable, and to be a sin. It is indeed true that envy causes a great sourness and unhappiness in people, but it is also a truth that only fools wallow in the adverse.

Great minds see opportunity in adversity, and linings of potential beauty in clouds of ugly desire. Envy can be a useful force; for those that are aimless in life, Envy is the clearest mirror that reveals the soul's truest desires. And hence, rather than repressing Envy, why not reach out and seize that which invokes Envy?

The parable of the scorpion and the frog tells us that we cannot deny our nature. Envy is ultimately an indicator of insufficiency, of lacking; our efforts to repress this sense of lacking serves only to enlarge our sense of insufficiency. Our nature demands a particular medicine and nothing else; only by obtaining the object of Envy can Envy be removed.

This is perhaps a sad truth, perhaps an ugly truth, but human nature is as such.