A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). A strobogrammatic prime is a strobogrammatic number that is also a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

, i.e., a number that is only

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

by one and itself (e.g., 11). It is a type of

ambigram An ambigram is a calligraphic composition of glyphs (letters, numbers, symbols or other shapes) that can yield different meanings depending on the orientation of observation. Most ambigrams are visual palindromes that rely on some kind of symmetry ...

, words and numbers that retain their meaning when viewed from a different perspective, such as

palindrome A palindrome (Help:IPA/English, /ˈpæl.ɪn.droʊm/) is a word, palindromic number, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as ''madam'' or ''racecar'', the date "Twosday, 02/02/2020" and th ...

Description

When written using standard characters (

ASCII ASCII ( ), an acronym for American Standard Code for Information Interchange, is a character encoding standard for representing a particular set of 95 (English language focused) printable character, printable and 33 control character, control c ...

), the numbers, 0, 1, 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other when rotated 180 degrees. In such a system, the first few strobogrammatic numbers are: 0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, 6699, 6889, 6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, ... The first few strobogrammatic primes are: :11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, ... The years 1881 and 1961 were the most recent strobogrammatic years; the next strobogrammatic year will be 6009. Although amateur aficionados of mathematics are quite interested in this concept, professional mathematicians generally are not. Like the concept of

repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book ''Recr ...

s and

palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...

s, the concept of strobogrammatic numbers is base-dependent (expanding to base-sixteen, for example, produces the additional symmetries of 3/E; some variants of

duodecimal The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is i ...

systems also have this and a symmetrical ''x''). Unlike palindromes, it is also font dependent. The concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.

Nonstandard systems

The strobogrammatic properties of a given number vary by

typeface A typeface (or font family) is a design of Letter (alphabet), letters, Numerical digit, numbers and other symbols, to be used in printing or for electronic display. Most typefaces include variations in size (e.g., 24 point), weight (e.g., light, ...

. For instance, in an ornate serif type, the numbers 2 and 7 may be rotations of each other; however, in a

seven-segment display A seven-segment display is a display device for Arabic numerals, less complex than a device that can show more characters such as dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic calculators, ...

emulator, this correspondence is lost, but 2 and 5 are both symmetrical. There are sets of glyphs for writing numbers in base 10, such as the

Devanagari Devanagari ( ; in script: , , ) is an Indic script used in the Indian subcontinent. It is a left-to-right abugida (a type of segmental Writing systems#Segmental systems: alphabets, writing system), based on the ancient ''Brāhmī script, Brā ...

and

Gurmukhi Gurmukhī ( , Shahmukhi: ) is an abugida developed from the Laṇḍā scripts, standardized and used by the second Sikh guru, Guru Angad (1504–1552). Commonly regarded as a Sikh script, Gurmukhi is used in Punjab, India as the official scrip ...

India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...

in which the numbers listed above are not strobogrammatic at all. In

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...

, given a glyph for 1 consisting of a single line without hooks or serifs and a sufficiently symmetric glyph for 0, the strobogrammatic numbers are the same as the palindromic numbers and also the same as the dihedral numbers. In particular, all Mersenne numbers are strobogrammatic in binary. Dihedral primes that do not use 2 or 5 are also strobogrammatic primes in binary. The natural numbers 0 and 1 are strobogrammatic in every base, with a sufficiently symmetric font, and they are the only natural numbers with this feature, since every natural number larger than one is represented by 10 in its own base. In

, the strobogrammatic numbers are (using inverted two and three for ten and eleven, respectively) :0, 1, 8, 11, 2↊, 3↋, 69, 88, 96, ↊2, ↋3, 101, 111, 181, 20↊, 21↊, 28↊, 30↋, 31↋, 38↋, 609, 619, 689, 808, 818, 888, 906, 916, 986, ↊02, ↊12, ↊82, ↋03, ↋13, ↋83, ... Examples of strobogrammatic primes in duodecimal are: :11, 3↋, 111, 181, 30↋, 12↊1, 13↋1, 311↋, 396↋, 3↊2↋, 11111, 11811, 130↋1, 16191, 18881, 1↋831, 3000↋, 3181↋, 328↊↋, 331↋↋, 338↋↋, 3689↋, 3818↋, 3888↋, ...

Upside down year

The most recent upside down year was 1961, or 2002 if the number 2 is included (in the case of seven-segment displays), and before that were sequentially 1881 and 1691, unless

leading zero A leading zero is any 0 digit that comes before the first nonzero digit in a number string in positional notation.. For example, James Bond's famous identifier, 007, has two leading zeros. Any zeros appearing to the left of the first non-zero dig ...

es are allowed to be arbitrarily added. In this case, 02020 would be the most recent upside down year. Before that were 1111 and 1001, and before that were 3-digit years, such as 986, 888, 689, 181, 101, etc. Using only the digits 0, 1, 6, 8 and 9, the next upside-down year will not occur until 6009. Allowing for the numbers 2, 5 and 7, the next such year will be 2112. '' Mad'' magazine parodied the upside down year in March 1961.

References

{{Use dmy dates, date=April 2019 Classes of prime numbers Integer sequences