Strobogrammatic number

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The number 619 is strobogrammatic. 619sign.JPG
The number 619 is strobogrammatic.

A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. [1] In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). [2] A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e., a number that is only divisible by one and itself (e.g., 11). [3] It is a type of ambigram, words and numbers that retain their meaning when viewed from a different perspective, such as palindromes. [4]



When written using standard characters (ASCII), 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, ... (sequence A000787 in the OEIS )

The first few strobogrammatic primes are:

11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, ... (sequence A007597 in the OEIS )

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 repunits 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. For instance, in an ornate serif type, the numbers 2 and 7 may be rotations of each other; however, in a seven-segment display 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 and Gurmukhi of India in which the numbers listed above are not strobogrammatic at all.

In binary, 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 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, and before that were sequentially 1881 and 1691. 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 (if leading zeroes are allowed to be arbitrarily added, 2020 can be made an upside down year by making it 02020).

Mad magazine parodied the upside down year in March 1961. [5] [6] [7]

Related Research Articles

A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.

222 is the natural number following 221 and preceding 223.

73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

In mathematics, a palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:

500 is the natural number following 499 and preceding 501.

700 is the natural number following 699 and preceding 701.

600 is the natural number following 599 and preceding 601.

800 is the natural number following 799 and preceding 801.

10,000 is the natural number following 9,999 and preceding 10,001.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

181 is the natural number following 180 and preceding 182.

10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.

100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.

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, and surface. The first few decimal dihedral primes are

202 is the natural number following 201 and preceding 203.

20,000 is the natural number that comes after 19,999 and before 20,001.

50,000 is the natural number that comes after 49,999 and before 50,001.

888 is the natural number following 887 and preceding 889.

A tetradicnumber, also known as a four-waynumber, is a number that remains the same when flipped back to front, flipped front to back, mirrored up-down, or flipped up-down. The only numbers that remain the same which turned up-side-down or mirrored are 0, 1, and 8, so a tetradic number is a palindromic number containing only 0, 1, and 8 as digits. The first few tetradic numbers are 1, 8, 11, 88, 101, 111, 181, 808, 818, ....


  1. "Strobogrammatic number". Encyclopædia Britannica . Retrieved 19 September 2021.
  2. Schaaf, William L. (1 March 2016) [1999]. "Number game". Encyclopedia Britannica. Retrieved 22 January 2017.
  3. Caldwell, Chris K. "The Prime Glossary: strobogrammatic". Retrieved 22 January 2017.
  4. Sloane, N. J. A. (ed.). "SequenceA000787(Strobogrammatic numbers: the same upside down)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 22 January 2017.
  5. Mad Magazine archival 'cover site'
  6. Mad Magazine, #61, March 1961. Upside Down Year. ASIN: B00ZJHXR4U