A Dudeney number is a positive integer where the sum of its decimal digits equals the cube root of the number.

## 123456789

The number (123…(*b*−1)) in base *b* has the property that when multiplied by any integer 1≤*k*≤*b*−1 which is coprime to *b*−1, its digits are permuted.

For example in base 10, **123456789**

∗2 = 246913578

∗4 = 493827156

∗5 = 617283945

∗7 = 864197523

∗8 = 987654312

Discover more intriguing facts about both useless and useful numbers on our website: https://archimedes-lab.org/numbers/Num1_69.html

## Arithmetic Mean ≥ Geometric Mean

A simple yet neat visual proof demonstrating that the arithmetic mean of two positive numbers ‘a’ and ‘b’ is always greater than or equal to their geometric mean, symbolically represented as (a+b)/2 ≥ √ab

## A Curious Constant

A palindromic number is an integer that remains the same when its digits are reversed. The sum of the reciprocals of all the palindromic numbers in the world converges to approximately *3.3703… *

## Analog Binary-to-Decimal Converter

This is a simple linkage-mechanism for converting binary numbers to decimal numbers.

## Amazing Double Cube

According to the Pythagorean theorem, adjacent cubes with side length 1 produce square roots of the first six natural numbers, as illustrated below:

Remarkably, by adding three extra cubes, we can extend the series of square roots of natural numbers up to √14. However, to obtain the square root of 7 using this method, we need to extend our analysis to a 4-dimensional world.

## Harshad Years

Harshad number is defined as an integer that is divisible by the sum of its digits.

Interestingly, the years 2022-2025 are Harshad numbers. It is worth noting that having more than two consecutive Harshad years is a rare occurrence. The last time it occurred was over 1000 years ago for years 1014-1017. The next time it is expected to occur after 1000+ years will be during the years 3030-3033.

## A Lucky Year!

7⁷ mod 7! = 2023 what a lucky year…

More amazing number patterns for getting 2023: https://community.wolfram.com/groups/-/m/t/2749012

## The Puzzling Ramanujan’s Magic Square

As you maybe know, a magic square is a square divided into smaller squares each containing a number, such that the figures in each vertical, horizontal, and diagonal row add up to the same value.

In this particular magic square by Ramanujan, fields of the same color add up to 139. The first row – highlighted in the bottom-right magic square – shows his date of birth.

## Summation Formulas

Some remarkable summation formulas…