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Automorphic Numbers And Other Fun Math Oddities

Fibonacci Spiral
Fibonacci Spiral – Credit: iShare

An automorphic number is one in which the square of the number as the same ending digits as the original number. These are also referred to as circular numbers due to their repetitive nature.

For example, there are only two automorphic numbers under 100 – the numbers 25 and 76. The square of 25 is 625, and the square of 76 is 5,776 . Some also consider 1 to be automorphic since it’s square is itself, as is its square root. The next three numbers that meet this criteria are 376, 625 and 9376. The number 625 is interesting in that it is an automorphic number created from another automorphic base. The square of 625 is 390,625 .

Another fascinating category of numbers are called “strobogrammatic”. Strobogrammatic numbers read the same after having been rotated 180 degrees (e.g., being flipped upside down). In the Arabic number system this works on the visualization that “0”, “1” and “8” are symmetrical through the horizontal axis and that “6” and “9” are flipped versions of the other. This also works for Roman numerals (e.g., “I” and “X” are horizontally symmetrical). For example the number 69 rotated 180 degrees becomes 96, and the number 101 is a mirror strobogrammatic number since it reads exactly the same after rotation.

Another class of numbers are called “Palindromic numbers”, which are conceptually the same as palindrome words (e.g., radar or rotor). These are numbers that remain the same after digits are reversed. The first palindromic numbers in the series are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, etc.

There are many more number categories including “repundits” (all the digits are “1′, for example 1 , 11, 111, 1111, etc. ) and the more well known”Fibonacci” (the next number in the series is the sum of the previous two, for example 1, 1, 2, 3, 5, 8, etc.).

A paper on creating number sequences by Tanya Khovanova is quite  interesting.

To test out a number sequence and identify it, (or to find out more about it), you can enter it at the On Line Encyclopedia of Integer Sequences (OEIS) .

Related Article – Complex Numbers Don’t Equate to Complex Math, It’s Imaginary

If you enjoy math puzzles and brain teasers, you should take a look at these books from Amazon:

Challenge Your Brain Math & Logic Puzzles ($8.80 – spiralbound)

Best Mathematical and Logic Puzzles ($4.95 – paperback)

The Stanford Mathematics Problem Book ($4.99 – paperback)

Entertaining Mathematical Puzzles ($6.99 – paperback)

The Moscow Puzzles: 359 Mathematical Recreations ($3.99 – paperback)

See here for more information on the pricing.

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