Fibonacci numbers and the golden ratio are not the only places where simple rules produce astonishing behaviour. This final lesson is a mathematical playground.
Kaprekar’s constant: 6174
Choose almost any four-digit number whose digits are not all the same. Rearrange its digits from largest to smallest and from smallest to largest, then subtract the smaller number from the larger. Repeat.
5432 − 2345 = 3087
8730 − 0378 = 8352
8532 − 2358 = 6174
7641 − 1467 = 6174
Choose a four-digit number with at least two different digits. How many steps does it take to reach 6174?
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The cyclic number 142857
From 1/7 we met the repeating block 142857. Multiplying it by 2, 3, 4, 5 or 6 rotates the same six digits.
142857 × 2 = 285714
× 3 = 428571
× 4 = 571428
× 5 = 714285
× 6 = 857142
Reveal answer
142857 × 7 = 999999. This is tied directly to the fact that 1/7 = 0.142857142857… .
Perfect numbers
A number is called perfect if its positive divisors smaller than itself add up to the number.
1 + 2 + 3 = 6
1 + 2 + 4 + 7 + 14 = 28
The next perfect numbers are 496 and 8128. Even perfect numbers are deeply connected with special prime numbers called Mersenne primes. Nobody knows whether an odd perfect number exists.
Find all the proper divisors of 496 and verify that they sum to 496.
Reveal answer
The proper divisors are 1, 2, 4, 8, 16, 31, 62, 124 and 248.
Their sum is 496.
Two giants: π and e
π ≈ 3.14159 begins with circles but appears throughout geometry, probability and infinite sums.
e ≈ 2.71828 appears naturally in continuous growth and can be approached by the expression (1 + 1/n)ⁿ as n gets larger.
Later mathematics brings them together with the imaginary number i in one of the most famous equations in mathematics:
eiπ + 1 = 0
You do not need to understand this equation yet. It is simply a glimpse of how ideas that first seem unrelated can eventually turn out to be connected.
What makes a number feel magical?
Usually not the number by itself. The magic comes from a simple definition producing unexpectedly rich connections.
- 6174 emerges from a simple digit algorithm.
- 142857 cycles because of the repeating decimal for 1/7.
- Perfect numbers link divisors and prime numbers.
- π and e begin in very different problems but appear throughout mathematics.
The best next question is not “what other magic numbers are there?” but “what simple rule can I experiment with next?”
Course challenge
Pick one idea from this course and investigate it further. You might test how quickly Fibonacci ratios approach φ, search for other cyclic decimals, record how many Kaprekar steps different starting numbers take, or hunt for patterns among perfect numbers.
Make a conjecture, test it, and try to explain why it might be true.