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Articles on numbers and patterns
In-depth, original explanations of Kaprekar’s constant and other approachable mathematical phenomena. Every claim is checked by calculation, and every article links to a tool where you can test it yourself.
Kaprekar’s constant and its relatives
Everything around 6174 and 495: the proof idea, the history, other digit lengths and the other number families Kaprekar named.
The algorithm
How Kaprekar’s routine works
The four steps of the routine, a full worked example and why leading zeroes and repdigits matter.
Number or nature?
Is 6174 found in nature?
Attractors, the hidden role of 9 and why 6174 belongs to base ten rather than to physics or biology.
Proof idea
Why does every number reach 6174?
The algebra behind the routine: why only 54 first results are possible and how they all funnel into one fixed point.
History
Who was D. R. Kaprekar?
A schoolteacher from Devlali who never held a research post, yet gave his name to one of the best-known number curiosities.
Beyond four digits
Kaprekar’s routine with 2, 5, 6 and 7 digits
Only three and four digits have a single destination. Other widths split into loops and competing fixed points.
A different Kaprekar idea
Kaprekar numbers: 45, 297 and split squares
45² = 2025 and 20 + 25 = 45. A separate Kaprekar discovery that is often confused with 6174.
Kaprekar’s other discoveries
Harshad numbers and self numbers
Two digit-sum ideas Kaprekar named: numbers that bring “joy” and numbers nobody can generate.
For teachers
Teaching 6174: a classroom lesson plan
A 45–60 minute lesson with a starter, group investigation, discussion questions and extension tasks.
More number patterns
Other famous iterations and digit curiosities, from the Fibonacci sequence to the unsolved Collatz conjecture.
Famous sequences
The Fibonacci sequence explained
Rabbits, ratios, the 60-step cycle of last digits and what the sequence really does in plants.
Separating fact from legend
The golden ratio: facts and myths
Where φ provably appears, and why the nautilus, the Parthenon and the “perfect face” do not belong on the list.
An unsolved problem
The Collatz conjecture (3n + 1)
A rule a child can follow that nobody has been able to prove always ends at 1.
Digit iterations
Happy numbers and the unhappy loop
Square the digits, add them, repeat. Every number ends at 1 or in one eight-number loop.
The power of 9
Digital roots and casting out nines
Why digit sums reveal remainders, how to check multiplication, and why every Kaprekar result is a multiple of 9.
Numbers in love with themselves
Narcissistic (Armstrong) numbers
153 = 1³ + 5³ + 3³. Why only 88 such numbers exist and the largest has 39 digits.
A classic number trick
The 1089 trick and why it works
Reverse, subtract, reverse, add: always 1089. The same multiples of 99 that power the 495 routine.