It was a gloomy evening and we were sipping some hot tea, peeking through the transparent window.
The day was so dull, that even the sun didn’t want to show up all day.
He hid behind the clouds, robbing us off our spirits.
The lacklustre day couldn’t get any more sluggish… we were craving for some entertainment.
So Little B decided to bombard us with one of his T-Nuts to cheer us up.
As trivial a problem that it seemed to be, we soon realised, nothing is ever so simple with Little B.
Hone your math skills
Little B started stating the problem.
“You are at the door of enlightenment.
Behind the door lies new realms of reality.
A world of peace, freedom and endless knowledge.
A world without constraints, concerns or constructs.
A world with no beginning and end.
An enticing utopia, freeing you from your chains.
You are just a step away from it.
A locked door standing between you and the paradise.
The door has a huge ‘XXX’ embossed on it.
The key to the door is the solution to a cryptic puzzle.”, he continued.
“Beside the door, are two crates.
The first crate, is full of spherical magic balls, etched with numbers.
While the second crate is empty.
The magic balls are all of the same diameter and shape.
The balls can freely move inside the crate.
They are perfectly smooth and glittering.
They vary in color and have different numbers etched on them as shown below.
Each ball has an unique number etched on it.
No two balls are engraved with the same number.
The Problem Statement
The cryptic puzzle he stated was as below:
“You need to pick, three balls from the first crate, no more, no less and place them in the second crate.
The constraint being, the sum of the carvings on the balls, should add to thirty,
XXX in roman, and thus the vignette on the door!
Three balls, No more, No less.
Summing to 30 (XXX)”
The Thought Train
Let’s begin cracking the T-Nut, like we always do, using the analogy of chewing on an unknown fruit.
This time however, we’ll follow the simplified guide, instead of the elaborate one.
VROOM VROOM! Let’s begin!
Step1 - Culling epicarp
I’m sure there was too much gibberish in that, so let me fish out the actual problem statement.
- The Naïve Problem Statement
It’s a simple math quiz.
Find x,y,z where
x + y + z = 30
Step2 - Biting
Visual representation is already provided. Let me think out loud to solve it.
Alright, 3 variables. I would need 3 equations at least.
Let me see if I can find the other two.
I find an inequation
x,y,z <=15
And they are all odd,
Any odd number can be represented as 2k+1, so
x = 2a + 1,
y = 2b + 1,
z = 2c + 1
where a,b,c are integers
It’s not much use to create more variables, is it?
Hey, wait a second. That’s an anomaly.
Sum of three odd numbers cannot be even!
Step3 - Chewing & Swallowing
Yup, surely. Sum of three odd cannot be even. Here’s the proof for non-believers.
x + y + z = 30
(2a + 1) + (2b + 1) + (2c + 1) = 30
2(a + b + c) + 3 = 30
2 (a + b + c + 1) + 1 = 30
if p = (a + b + c + 1),
2p + 1 = 30
(2p + 1) represents an odd number.
(2p + 1) cannot be 30 when p is an integer!!!
Odd cannot be equal to even
So, NO SOLUTION
So, we cracked the nut and were flaunting our brilliance with Little B.
The Mockery
As we were basking in the glory of cracking the t-nut, all Little B could manage was a satirical grin.
Infuriated by his disdainful visage, we enquired as to what was so funny to smile?
We asked him,
“Is accepting that we could crack your T-Nut, with much ease, such a bitter pill to swallow?”
To this, he just stood up and started walking away. He paused momentarily and with his gaze directed away from us, without even turning towards our side, he mocked,
“You stand at the doors of heaven, with a free ticket to freedom.
All you need to do is choose three freaking balls that’ll redefine your life for ever.
And what do you end up doing?
Laugh at the door, taunting that the question is wrong.
And you even have the audacity to rejoice the moment?
The trouble with all of us, is that, we don’t know to define victory.
We don’t even understand our battles, letgo choosing them.
Bravo Guys!!! Have fun!!” he said and faded into the twilight.
More than the way he scorned at the us, the depth of his ridicule and the utter hopelessness of our approach, highlighted in his derision, broke us apart.
So true… when you think of it…
I’m a step away from paradise.
And I’m letting a freaking elementary math problem block my way?
No way, time to go full throttle, I thought to myself.
I jumped at the problem again with renewed resolve.
Bite, Chew & Swallow
However I look at it, the sum of three odd numbers cannot turn even.
There is nothing more in the problem to extract as well. The crate with balls is there for full view.
After a sleepless night of futile effort, I ran to Little B to understand the solution.
I was heart-broken but I needed the solution.
Am I so bad at math?
What am I missing?
The Solution
Little B smiled at our inanity and picked three balls as below to show the solution.
“Perspective my friend, perspective”, he said.
As soon as I looked at it, the whole world came crashing at me.
“Come on!!! How the heck did I miss that?”, I said to myself.
It was impossible to digest my folly… my idiocy.
Huh!!! Unbearable!!!
Where did I lose track, I wondered and re-read the key aspect in the question that I had missed.
“The balls can freely move inside the crate”
With hands in his pocket, Little B stood up sluggishly, and started walking away, muttering,
“All you need to do. is let go, and look at the problem from a different angle”,
And faded into the distance, not even bothering to look by my side.
I looked out of the window, as the sun was slowly slipping out of the cloud.