The Mathematical Black Hole: Why Do We Always End Up at FOUR?

Imagine a number that acts like a black hole. No matter where you start, if you follow a simple rule, you'll eventually get trapped at FOUR!

This fascinating mathematical trick is based on counting the letters in number names. Let's explore how it works with two examples.

How Does the Mathematical Black Hole Work?

The rule is simple:

1. Choose any English word.
2. Count the number of letters in the word.
3. Spell out that number in English.
4. Count the letters in the new word.
5. Repeat the process until you reach FOUR.

Let's see what happens!

Example 1: Starting with MATHEMATICS

Let's begin with the word MATHEMATICS.

Step 1: MATHEMATICS contains 11 letters.

Step 2: Spell out 11: ELEVEN.

Step 3: ELEVEN contains 6 letters. Spell out 6: SIX.

Step 4: SIX contains 3 letters. Spell out 3: THREE.

Step 5: THREE contains 5 letters. Spell out 5: FIVE.

Step 6: FIVE contains 4 letters. Spell out 4: FOUR.

Now, here's the amazing part!

FOUR contains exactly 4 letters. Therefore, the process continues forever:

FOUR → FOUR → FOUR → FOUR → ...

We've reached the mathematical black hole!

Example 2: Starting with COMPUTER

Let's try another word: COMPUTER.

Step 1: COMPUTER contains 8 letters.

Step 2: Spell out 8: EIGHT.

Step 3: EIGHT contains 5 letters. Spell out 5: FIVE.

Step 4: FIVE contains 4 letters. Spell out 4: FOUR.

And once again, we're trapped!

COMPUTER → EIGHT → FIVE → FOUR → FOUR → ...

No matter how many times we repeat the process, we keep getting FOUR.

Why Is FOUR Called a Mathematical Black Hole?

A black hole is an object from which nothing can escape. In this mathematical puzzle, FOUR behaves similarly.

Once we reach FOUR, counting its letters gives us 4 again. This means the process repeats indefinitely.

In mathematics, this is an example of a fixed point: a value that remains unchanged when a particular operation is applied to it.

Here, the operation is counting the letters in the English name of a number, and FOUR is its fixed point.

Homework Challenge

Now it's your turn!

Start with the word EXTRAORDINARY.

Count its letters, spell out the resulting number, and continue following the same rule.

Will you eventually reach FOUR?

Try it yourself and share your answer in the comments!

Conclusion

This simple puzzle shows how an ordinary counting process can lead to a fascinating mathematical result. Starting with different words, we can follow a chain of number names until we reach FOUR, where the process becomes a never-ending loop.

Can you find a word that escapes this mathematical black hole?

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Alok Pradhan

28.09.2026

Cuttack.

Divisibility tricks for 7, 11, 13, 17, 19 & 23 — (easy rules + worked examples)

Introduction

When you want to check whether a large number is divisible by a given small prime (or composite) without long division, divisibility tricks are handy. Many rules are iterative: they transform the number into a smaller one that has the same remainder mod the divisor. Below are simple, reliable rules for 7, 11, 13, 17, 19, and 23 — each with clear examples you can easily understand.

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1) Divisibility by 7

Rule (one common version): Remove the last digit, double it, and subtract that from the remaining truncated number. Repeat until you get a small number. If the result is divisible by 7, the original number is divisible by 7.

Why (brief): If number = 10×(rest) + lastDigit. Since 10 ≡ 3 (mod 7), you can transform to an equivalent smaller expression; the “subtract double the last digit” method works by producing an equivalent remainder modulo 7.

Example: Test 20,203

1. Split: rest = 2020, last = 3. Compute 2020 − 2×3 = 2020 − 6 = 2014.

2. Repeat: rest = 201, last = 4 → 201 − 2×4 = 201 − 8 = 193.

3. Repeat: rest = 19, last = 3 → 19 − 2×3 = 19 − 6 = 13.
   13 is not divisible by 7 → 20,203 is not divisible by 7.

Another quick example: 203 → 20 − 2×3 = 20 − 6 = 14 → 14 is divisible by 7 → 203 is divisible by 7.

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2) Divisibility by 11

Rule (standard): Compute the alternating sum of digits (sum of digits in odd positions − sum of digits in even positions). If that result is a multiple of 11 (including 0 or ±11, ±22, …), the original number is divisible by 11.

Example: Test 27,528
Digits from left: 2 7 5 2 8
Alternating sum = (2 − 7 + 5 − 2 + 8) = 6 → 6 is not a multiple of 11 → 27,528 is not divisible by 11.

Example 2: 34,287 → (3 − 4 + 2 − 8 + 7) = 0 → 0 is divisible by 11 → 34,287 is divisible by 11.

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3) Divisibility by 13

Rule (useful iterative rule): Remove the last digit, multiply it by 4, and add that to the truncated number. Repeat until small. If the result is divisible by 13, the original number is divisible by 13.

Example: Test 351

1. rest = 35, last = 1 → 35 + 4×1 = 35 + 4 = 39.

2. 39 is divisible by 13 → 351 is divisible by 13 (indeed 13×27 = 351).

Another example: Test 2,226

1. rest = 222, last = 6 → 222 + 4×6 = 222 + 24 = 246.

2. rest = 24, last = 6 → 24 + 4×6 = 24 + 24 = 48.

3. 48 is not divisible by 13 → 2,226 is not divisible by 13.

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4) Divisibility by 17

Rule (one easy form): Remove the last digit, multiply it by 5, and subtract from the truncated number. Repeat. If the final number is divisible by 17, the original number is too.

(We use: new = rest − 5×last)

Example: Test 289

1. rest = 28, last = 9 → 28 − 5×9 = 28 − 45 = −17.
   −17 is divisible by 17 → 289 is divisible by 17 (indeed 17×17 = 289).

Another quick one: 1,533 → rest 153, last 3 → 153 − 15 = 138 → 13 − 5×8? (we continue) rest 13, last 8 → 13 − 40 = −27 (not divisible by 17) → 1,533 is not divisible by 17.

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5) Divisibility by 19

Rule (simple): Remove the last digit, multiply it by 2, and add to the truncated number. Repeat. If the final result is divisible by 19, so is the original number.

(new = rest + 2×last)

Example: Test 133
rest = 13, last = 3 → 13 + 2×3 = 13 + 6 = 19 → divisible → 133 is divisible by 19.

Another: 1,902
rest = 190, last = 2 → 190 + 4 = 194.
rest = 19, last = 4 → 19 + 8 = 27 → not divisible by 19 → 1,902 not divisible by 19.

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6) Divisibility by 23

Rule (one version): Remove the last digit, multiply it by 7, and add to the truncated number. Repeat. If the result is divisible by 23, the original number is divisible by 23.

(new = rest + 7×last)

Example: Test 115
rest = 11, last = 5 → 11 + 7×5 = 11 + 35 = 46 → 46 = 2×23 → 115 is divisible by 23.

Another: Test 5,842
rest = 584, last = 2 → 584 + 14 = 598.
rest = 59, last = 8 → 59 + 56 = 115 → 115 is divisible by 23 → 5,842 is divisible by 23.

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Practice problems

Try these quickly (work with the given rules); answers below.

1. Is 1,234 divisible by 7?

2. Is 11,891 divisible by 11?

3. Is 1,014 divisible by 13?

4. Is 5,102 divisible by 17?

5. Is 2,285 divisible by 19?

6. Is 4,602 divisible by 23?

Answers (work shown briefly)

1. 1,234 → (for 7) 123 − 2×4 = 123 − 8 = 115 → 11 − 2×5 = 11 − 10 = 1 → not divisible → No.

2. 11,891 → alternating sum = 1 − 1 + 8 − 9 + 1 = 0 → divisible → Yes.

3. 1,014 → for 13: 101 + 4×4 = 101 + 16 = 117 → 11 + 4×7 = 11 + 28 = 39 → 39 = 3×13 → Yes.

4. 5,102 → for 17: 510 − 5×2 = 510 − 10 = 500 → 50 − 5×0 = 50 → 5 − 5×0 = 5 → not divisible → No.

5. 2,285 → for 19: 228 + 2×5 = 228 + 10 = 238 → 23 + 2×8 = 23 + 16 = 39 → 39 not multiple of 19 → No. (39 = 19×2 + 1)

6. 4,602 → for 23: 460 + 7×2 = 460 + 14 = 474 → 47 + 7×4 = 47 + 28 = 75 → not divisible → No.

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Quick tips & final notes

• Apply the operation repeatedly until the number is small enough to check by eye. You may need to work with negatives — if the small final result is a multiple of the divisor, the original number is divisible.

• These rules arise because of how 10 behaves modulo the divisor; each rule transforms 10×rest + last into an equivalent (mod divisor) smaller expression.

• If you prefer, you can always do regular division or use a calculator — but these tricks are fast in head-calculation and great for contests and interviews.

Fascinating Ancient Numbers : Greek Numerals (Attic and Ionian)

The history of numbers is a fascinating journey through human civilization.

Early Beginnings: In prehistoric times, the earliest evidence of counting came from tally marks on bones and stones, dating back to around 35,000 BCE. The Ishango Bone, discovered in Africa, is a notable example. Also, humans likely used physical objects like pebbles, sticks, or knots in a string to keep track of numbers.

Ancient Civilizations: Around 3000 BCE, the Sumerians developed one of the earliest number systems, using a base-60 (sexagesimal) system. The Babylonians inherited and expanded this system, influencing their mathematics and astronomy. It inspired ancient Egyptians to create a decimal (base-10) system around 3000 BCE. Their hieroglyphs represented numbers and were used in administrative and architectural contexts.

Greeks: The Greeks used a variety of systems, including the Attic and the later alphabetic (Ionian) numerals, which assigned values to letters of their alphabet.

Attic: 

• 1: | (a vertical stroke)
• 5: Π (the first letter of "pente," meaning five)
• 10: Δ (the first letter of "deka," meaning ten)
• 50: ΠΔ (a combination of the symbols for 5 and 10)
• 100: Η (the first letter of "hekaton," meaning hundred)
• 500: ΠΗ (a combination of the symbols for 5 and 100)
• 1,000: Χ (the first letter of "chilioi," meaning thousand)
• 5,000: ΠΧ (a combination of the symbols for 5 and 1,000)
• 10,000: Μ (the first letter of "myrioi," meaning ten thousand)
  
  

    This system was additive, meaning that the values of the symbols were added together to form numbers. This system was additive, meaning that the values of the symbols were added together to form numbers. 

For example:  49 = 40 + 9 = ΔΔΔΔ + ΠΙΙΙΙ = ΔΔΔΔΠΙΙΙΙ

Ionian (Alphabetic):

• 1-9:
• 1: Α (alpha)
• 2: Β (beta)
• 3: Γ (gamma)
• 4: Δ (delta)
• 5: Ε (epsilon)
• 6: Ϛ (stigma)
• 7: Ζ (zeta)
• 8: Η (eta)
• 9: Θ (theta)
• 10-90:
• 10: Ι (iota)
• 20: Κ (kappa)
• 30: Λ (lambda)
• 40: Μ (mu)
• 50: Ν (nu)
• 60: Ξ (xi)
• 70: Ο (omicron)
• 80: Π (pi)
• 90: ϙ (koppa)
• 100-900:
• 100: Ρ (rho)
• 200: Σ (sigma)
• 300: Τ (tau)
• 400: Υ (upsilon)
• 500: Φ (phi)
• 600: Χ (chi)
• 700: Ψ (psi)
• 800: Ω (omega)
• 900: ϡ (sampi)

In this system, numbers were written by combining these letters. 

For example: the number 241 = ΣΜΑ (200 + 40 + 1).

The Ionian system was more efficient and flexible than the Attic system, leading to its widespread adoption throughout the Greek world. Thereafter, the Roman numeral system, developed around 500 BCE, used combinations of letters from the Latin alphabet (I, V, X, L, C, D, M) to represent numbers.

        The Attic and Ionian numeral systems played a crucial role in various aspects of ancient Greek life, contributing to their economic prosperity, scientific advancements, cultural achievements, and administrative efficiency.

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Credits:

• OpenAI. (2024). ChatGPT [Large language model]. https://chatgpt.com/c/de7a198b-bf21-4493-84d4-35227eb07f58
• Attic numerals. (2023, June 1). In Wikipedia. https://en.wikipedia.org/wiki/Attic_numerals
• Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media, 2013.

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Thank You

Alok Pradhan

30.07.2024

What is TRP and how it's calculated?

What is TRP?

TRP stands for Television Rating Point / Target Rating Point.

It shows the rating of TV channels and which shows are viewed the most.

Who calculates TRP?

TRP is calculated by a industry body called Broadcast Audience Research Council (BARC).

Before BARC, bam was the rating agency for measuring TRP.

BARC publishes the weekly rankings for channels.

How TRP is calculated?

To calculate TRP a device is used called peoplemeter.

Peoplemeter is installed with TV, it records how much time the channels are viewed.

As it is not possible to install this device in each television of the country, a sample size of 44000+ houses are selected.

It installs peoplemeter in the selective houses in different region of the country and each week measurement is taken from these households.

To install peoplemeter BARC has given contract to Hansa research.

On the basis of the sample size a statistical data, rating is given to the channels.

This Rating shows which channels are popular or seen most throughout the country.

Example:

Suppose there are  2 channels named A&B.

A is seen by 6 peoples for 10 minutes by each.

B is seen by 4 people for 20 minutes by each.

Total time watched for A = 10*6 = 60 min for B=20*4=80 min.

So B's Rating is higher than that of A.

Why TRP is important:

•Higher TRP attracts advertisers as their ad will reach more audience. It enables advertisers to determine which channels are viewed most and which show is popular.

•TRP rating enables broadcaster to charge advertiser accordingly. It means higher TRP of a channels implies higher earning through advertisement.

Loophole of TRP:

It doesn't matter how many people are watching a particular channel, but how much time a channels is seen matters. So isn't it easy to manipulate the TRP by luring the viewers where peoplemeter are used.

There are about 20 Crore TV sets and about 80 Crore people watch TV in India.

Only 44000 peoplemeter or TRP meter are installed which is only 0.00022 percent or 22 peoplemeter for  every 1 lakh TV. It means 22 people decides what 1 lakh people are watching.

It's not that each of the  44000 peoplemetre measures all type of channels of a TV set.

•23760 (54%) device are used for HIndi entertainment only.

•Only 8000 (20%) devices are used Hindi News channels.

i,e. 4 TV user decides what 1 lakh TV users are watching.

•Only 700 devices are used for English News channels.

i,e. 35 TV for every 1 Crore. 35 people decides what 1 Crore people are watching.

What do you think?

Is this the proper way of measuring TRP? 

The likes and dislikes of 20,00,00,000 TV users are being measured by only 44,000 TV users.

TRP can be easily manipulated by bribing the TV users where peoplemeter is installed and a channel's rating can be easily increased.

Comment below what do you think about it and in your opinion how TRP can be calculated authentically.

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Light Year | Distance

A light year is a unit of distance. 

It is defined as the distance that light travels in one year. 

 

In our daily lives, we typically use units such as centimeters, meters, kilometers, miles, inches, and feet to measure distances between two points. However, these units are not suitable for measuring the vast distances between astronomical objects like planets, stars, and galaxies. Therefore, astronomers use the light year as a more appropriate measurement.

1 Light Year = 9.4607×10¹² km 

•Light travels at the rate of 3×10⁸ m/sec in vacuum.

•Light travels at the rate of 2.2×10⁸ m/sec in water.

•Light travels at the rate of 2×10⁸ m/sec in glass.

DO YOU KNOW?

•Milky Way galaxy is about 1,000,000,000,000,000,000 km (about 100,000 light years).

There are billions of galaxy in the observable universe.

•Nearest galaxy to our Milky Way galaxy is Andromeda Galaxy which is 2.3 million light years away.

Just imagine how long the distance is and how big our universe.

•The radius of observable universe is 46.508 billion light years.

Think how our science has developed that we can study the planets,stars,galaxies up to this huge distance.

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If you know more interesting facts like this then comment below.

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How a day, a month and a year is defined?

DAY:

A Day is defined as the time taken by earth to complete one rotation around its axis.

Earth takes approximately 23 hour and 56 minutes to complete one rotation around its axis. So one day of earth is taken as 24 hours. 

Now let's see how much time planets in our solar system take to complete one rotation around its axis.

MONTH:

Generally a  Month is defined as time taken by MOON to complete one revolution around earth which is approximated to 30 days.

YEAR:

A Year is defined as the time taken by earth to complete one revolution around Sun.

Earth takes 365.242  days to complete one revolution around sun.

Every year is considered as 365 days and to adjust 0.242 day which is approximately equal to 0.25 day, in every fourth year one day is added which is called Leap Year.

This extra day is added to the month February, which becomes 29 days. 

see, how long it takes by the planets of our solar system to complete one revolution around sun (in earth days)

Thank You...

For any query or suggestion comment below

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PIN Code and what does its each digit mean

 PIN stands for Postal Index Number.

A PIN refers to a six digit code in the Indian Postal System used by India Post.

HISTORY:

India Post is a government-operated postal system that was established in 1854, this is the largest postal service or most widely distributed postal service in the world.

It had some limitations like manual sorting of posts and delivery of posts by eliminating confusion over names of similar places, and incorrect delivery addresses.

To simplify this PIN system was launched on 15th August 1972 by the Ministry of Communication. This system divided India into 9 different zones and each zones into many sub-zones. The first eight zones i,e. digits 1 to 8 are divided geographically and the digit 9 is reserved for Army Postal.

PIN STRUCTURE

Each digit of PIN code is assigned to a specific region/location. 

First digit of the PIN refers to the zone.

• 1- Delhi, Haryana, Punjab, Himachal Pradesh, Jammu & Kashmir, Ladakh, Chandigarh
• 2- Uttar Pradesh, Uttarakhand
• 3- Rajasthan, Gujarat, Daman and Diu, Dadra and Nagar Haveli
• 4- Maharastra, Goa, Madhya Pradesh, Chhatisgarh
• 5- Telengana, Andhra Pradesh, Karnataka
• 6- Tamilnadu, Kerala, Puducherry, Kerala
• 7- West Bengal, Odisha, Nagaland, Arunachal Pradesh, Tripura, Mizoram, Tripura, Meghalaya, Assam, Sikkim, Andaman & Nichobar Island.
• 8- Bihar, Jharlhand
• 9- Army Post Office(APO), Field Post Office(FPO)

Army Postal Service is a government operated postal service for the delivery of mail of army.

Field Post Office is set up at the time of war or military exercise.

Second digit indicate the sub-zone.

Second digit along with the first digit indiate a sub-zone or a circle.

• 11 - Delhi
• 12,13 - Haryana
• 14,15 - Punjab
• 16 - Chhatisgarh
• 17 - Himachal Pradesh
• 18,19 - Jammu & Kashmir, Ladakh
• 20 to 28 - Uttar Pradesh, Uttarakhand
• 30 to 34 - Rajasthan

• 36 to 39 - Gujurat
• 40 to 44 - Maharstra
• 45 to 49 - Chhatisgarh, Madhya Pradesh
• 50 to 53 - Andhra Pradesh, Telengana
• 56 to 59 - Karnataka
• 60 to 64 - Tamil Nadu
• 67 to 69 - Kerala
• 70 to 74 - West Bengal
• 75 to 77 - Odisha
• 78 - Assam
• 79 - North Eastern State
• 80 to 85 - Bihar, Jharkhand
• 90 to 99 - Army Postal Service

Third digit refers to a geographical region (excluding in the case of functional zone of Army)  called sorting District i,e. headquartered at the main post office of the largest city of the region.

Fourth digit represent the route on which a post office is located. The digit 0 is for offices which are nearer to the sorting district.

Fifth and Sixth digit together represent a specific Post Office starting from 01.

Thank You...

For any query or suggestion comment below 

                                                                  

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