The Jueteng Combination

Two weeks after it was posted, the puzzle in my article Amusing Challenge 2 remains unresolved except for the cryptic comment posted by Carlo Chan which hinted on the correct answer but left out the rationale. Knowing Carlo, he either is too busy attending at his daytime job (He is a Structural Engineer at BG&E Consulting) or he considers the puzzle too simple as to be self-evident. A third but unlikely explanation is that he is one of them - ET's. hehehe. So here's how I would have solved it...

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The Jueteng Combination

by icarus

A lot of people were thrown off at their attempt at making sense of the puzzle because they made wrong assumptions. Most start their solutions using ratios and scaling the numbers to fit the given sums and products. But it doesn’t work that way and invariably, they end up with contradictions and inconsistencies.

A subtle clue which most people miss or dismiss as inconsequential is the number of fingers in those extra terrestrials: four fingers in each hand.

Have you ever wondered why we usually, almost intuitively, group things by ten or multiples of ten? Or why we have ten numerical symbols, 0 to 9? Yes, you got it right - it’s because we have ten fingers in our hands! Our number systems evolved from our primitive way of counting with our fingers. But as the homo sapien advances, the need to count beyond ten also became a real challenge. Man ran out of fingers to use and he cant possibly have unique symbols to represent the infinitely large number of integers. Thus, place values in the numeric system were invented. The 5 in 25 is totally different from the 5 in 524 because they occupy different locations, although they have exactly the same symbol. Using this concept, we actually meant:

524 = 5 x 10^2 + 2 x 10^1 + 4 x 10^0 = 5 x 100 + 2 x 10 + 4 x 1 = 500 + 20 + 4
25 = 2 x 10^1 + 5 x 10^0 = 2 x 10 + 5 x 1 = 20 + 5

To us in the present age, this idea of place values is not given a second glance - no big deal -as if it is a natural phenomenon - but it is not! It was one of the most remarkable concepts ever developed by man. If you don’t believe me, try dividing 2364 by 8 using the less evolved Roman Numerals!

Which bring us back to the ET’s with 8 fingers in their hands. We can hypothesize that they will also be grouping things by eight, not ten, as a matter of natural course. As such, they will only use 8 symbols, ( 0 to 7). And they will use powers of 8 to define their place values much like we use powers of 10 (and much like the electrical logic circuit which uses powers of two because it only got two states, on or off representing one or zero)

Thus, if we test our theory on the given sum we prove that:

765 + 45
= (7 x 8^2 + 6 x 8^1 + 5 x 8^0) + (4 x 8^1 + 5 x 8^0)
= 501 + 37
= 538 in our decimal system.

1,032
(1 x 8^3 + 0 x 8^2 + 3 x 8^1 + 2 x 8^0)
= 538 also, in our decimal system.

2 x 63
= (2 x 8^0) x (6 x8^1 + 3 x 8^0)
= 2 x 51
= 102 in our decimal system.

146
= (1 x 8^2 + 4 x 8^1 + 6 x 8^0)
= 102 also,in our decimal system.

Thus, it is verified that the numbers were actually written in base 8.

In same manner, you can arrive at the winning combination being sought by my friend, Pidol which is:

16 + 3 = 21

72 - 54 = 16

both in the octal system.

Anunsiyo: Naparagsit nga Pandek!

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hehehe...

-ise.