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 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 because the only ways ...

, i.e., a number that is only divisible by one and itself (e.g., 11). It is a type of

ambigram An ambigram is a calligraphic design that has several interpretations as written. The term was coined by Douglas Hofstadter in 1983. Most often, ambigrams appear as visually symmetrical words. When flipped, they remain unchanged, or they mutate ...

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

Description

When written using standard characters (

ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...

), 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 ''Recreat ...

s and palindromic numbers, 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 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 the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font. There are thousands o ...

. 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 form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic ...

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 ( ; , , Sanskrit pronunciation: ), also called Nagari (),Kathleen Kuiper (2010), The Culture of India, New York: The Rosen Publishing Group, , page 83 is a left-to-right abugida (a type of segmental writing system), based on the ...

and

Gurmukhi Gurmukhī ( pa, ਗੁਰਮੁਖੀ, , Shahmukhi: ) is an abugida developed from the Laṇḍā scripts, standardized and used by the second Sikh guru, Guru Angad (1504–1552). It is used by Punjabi Sikhs to write the language, commonly ...

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

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 digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...

, 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 In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

are strobogrammatic in binary.

Dihedral prime A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside down), and surface (actual display or reflectio ...

s 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 duodecimal, the strobogrammatic numbers are (using inverted two and three for ten and eleven, respectively) :0, 1, 8, 11, 2ᘔ, 3Ɛ, 69, 88, 96, ᘔ2, Ɛ3, 101, 111, 181, 20ᘔ, 21ᘔ, 28ᘔ, 30Ɛ, 31Ɛ, 38Ɛ, 609, 619, 689, 808, 818, 888, 906, 916, 986, ᘔ02, ᘔ12, ᘔ82, Ɛ03, Ɛ13, Ɛ83, ... Examples of strobogrammatic primes in duodecimal are: :11, 3Ɛ, 111, 181, 30Ɛ, 12ᘔ1, 13Ɛ1, 311Ɛ, 396Ɛ, 3ᘔ2Ɛ, 11111, 11811, 130Ɛ1, 16191, 18881, 1Ɛ831, 3000Ɛ, 3181Ɛ, 328ᘔƐ, 331ƐƐ, 338ƐƐ, 3689Ɛ, 3818Ɛ, 3888Ɛ, ...

Upside down year

The most recent upside down year was 1961, or 2002 if the number 2 is included, and before that were sequentially 1881 and 1691, unless leading zeroes 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 While the future cannot be predicted with certainty, present understanding in various scientific fields allows for the prediction of some far-future events, if only in the broadest outline. These fields include astrophysics, which studies ho ...

. 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

External links

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