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]

Contents

Description

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, 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, 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. 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. [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:

11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.

111 is the natural number following 110 and preceding 112.

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.

101 is the natural number following 100 and preceding 102.

500 is the natural number following 499 and preceding 501.

600 is the natural number following 599 and preceding 601.

800 is the natural number following 799 and preceding 801.

900 is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10 it is a Harshad number. It is also the first number to be the square of a sphenic number.

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

<span class="mw-page-title-main">1,000,000</span> Natural number

1,000,000, or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, "thousand", plus the augmentative suffix -one.

A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

181 is the natural number following 180 and preceding 182.

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

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.

90,000 is the natural number following 89,999 and preceding 90,001. It is the sum of the cubes of the first 24 positive integers, and is the square of 300.

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, ....

References

  1. ↑ "Strobogrammatic number". Encyclopædia Britannica . Archived from the original on 21 September 2021. Retrieved 19 September 2021.
  2. ↑ Schaaf, William L. (1 March 2016) [1999]. "Number game". Encyclopedia Britannica. Archived from the original on 2 February 2017. Retrieved 22 January 2017.
  3. ↑ Caldwell, Chris K. "The Prime Glossary: strobogrammatic". primes.utm.edu. Archived from the original on 8 January 2017. 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'". Archived from the original on 15 November 2020. Retrieved 12 September 2022.
  6. ↑ "Mad Magazine, #61, March 1961. Upside Down Year. ASIN: B00ZJHXR4U". Archived from the original on 19 February 2020. Retrieved 12 September 2022.
  7. ↑ "MAD MAGAZINE MARCH 1961 #61 UPSIDE-DOWN YEAR SPY VS SPY. WorthPoint". Archived from the original on 5 February 2020. Retrieved 12 September 2022.

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