WHAT IS A CAPICÚA NUMBER? PROPERTIES AND EXAMPLES - MATHS - 2025

A capicúa number is the one that is read the same from left to right, as from right to left. The Royal Spanish Academy reserves the word palindrome when it is a phrase, sentence or word that reads the same in both senses.

Examples of capicúa numbers are: 2002, 11, 121, 34543. And as examples of palindromic words are: eye, ana, alone. There you see Seville, it is a palindromic phrase.

Figure 1. Some capicúa numbers. Source: F. Zapata.

The word capicúa originates from the Catalan: cap i cua which means head and tail, and is applied exclusively to numerical figures that are read the same in both directions.

These figures are also called palindrome numbers, although "palindrome" is mostly used in reference to words or phrases that are read the same backwards and forwards.

In the English language there is no word capicúa, so these types of numbers are called "palindromic numbers".

Properties of capicúa numbers

1.- Every one-digit number is considered capicúa: 0 is capicúa, 5 is capicúa, and so on.

2.- Capicúa numbers form a succession of digits, in such a way that the first and last are equal, the second and penultimate are equal and so on.

In 1234321, the first equals the last, the second equals the penultimate, the third equals the penultimate, and the middle equals itself.

3.- If the number of digits in a capicúa number is odd, then the central digit is the center of symmetry, so that the digits that are the same distance from the central digit are equal.

Consider 73137, in this number 1 is the central digit and center of symmetry, 3 is one distance from the central digit and 7 is two distance from the central digit in both directions.

4.- If the number of digits is even, the center of symmetry is the space that divides the same number of digits on the right as on the left.

The digits that are the same distance (or steps) from this center are necessarily equal.

For example: 731137, in this number the center of symmetry is in the space between the two numbers 1. The other digits are located symmetrically on either side of this center.

5.- A capicúa number with an even number of digits is divisible by 11:

6666 ÷ 11 = 606

3223 ÷ 11 = 293

6.- A capicúa number can be obtained by adding any number with its reverse successive times until obtaining a capicúa:

45 + 54 = 99

75 + 57 = 132 → 132 + 231 = 363

913 + 319 = 1232 → 1232 + 2321 = 3553

7.- Between 10 and 99 there are 9 capicúa numbers. Between 100 and 999 there are 93 capicuas, among which is 666, known as the number of the beast, associated with the devil.

Capicúa successions

8.- A sequence with a finite number of terms is capicous if the equidistant terms are equal: {a ₀, a ₁,…, a _i,…, a _n-1, a _n } is capicua if a _i = a _{n -1}.

9.- When the sum of the n elements of a geometric progression, which starts at 1 and has a ratio r, is raised to a natural exponent, the coefficients of its development generate a capicúa sequence.

For example, consider the geometric progression of ratio r and four elements raised to the exponent 3:

(1 + r¹ + r² + r³ + r⁴) ³

The development of this expression is

r¹² + 3r¹¹ + 6r¹⁰ + 10r⁹ + 15r⁸ + 18r⁷ + 19 r⁶ + 18r⁵ + 15r⁴ + 10r³ + 6r² + 3r + 1

If the sequence of the coefficients is formed in an ordered way, we have

{1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1}

What is a capicúa succession.

Capicua dates

On Sunday, February 2, 2020, it was baptized as the “palindrome day”, during that week the hashtag #PalindromeDay was a trend on the networks.

The reason is that this date is capicúa in the day / month / year format, in the month / day / year format, or in the year / month / day format:

02/02/2020

2020/02/02

The date before this, on which the format-independent capicúa date coincided, occurred 908 years two months and 22 days before, and was November 11, 1111, that is, 11/11/1111.

The next date capicúa independent of the format of 101 years, 10 months and 10 days after the designated day: 12/12/2121

In the day / month / year format, the capicúa before #PalindromeDay was 10 years ago and one day: 02/01/2010, while the next one will be 1 year and 10 days later: 02/12/2021.

For believers in superstition, capicua dates often represent doom or good luck. In Ancient Greece it was believed that these dates could be apocalyptic.

In the same way, lottery tickets with capicúa numbers are considered good luck and fans collect them.

Figure 2. Capicúa tickets are considered good luck and are collectibles. Source: Wikimedia Commons.

Capicúa challenges

Guess the ticket number

The lucky ticket has a capicúa figure. The sum of its five digits gives the same result as if they are multiplied. The first digit is the age of my little sister, the next two the age of the middle sister, and the last two are the age of the older sister, which takes more than a year to the middle one.

Guess what number i am

I have four figures, I am a capicúa, some see me as if it were a 9 and between 2 and 10 I only have a divisor.

a.- 1111, b.- 1001, c.- 2002, d.- 1221, e.- 2112

How many capicuas are there between 100 and 200?

a.- 5, b.- 7, c.- 10, d.- 52, e.- 89

Lychrel numbers

Previously it was said that a capicúa number can be obtained by adding any number with its reverse successive times until obtaining a capicúa.

It seems that this rule has its exceptions for certain numbers called Lychrel numbers, of which the existence of some has not been proven, perhaps because a very large number of tests must be done before obtaining a capicúa. A candidate for Lychrel's number is 196.

As a challenge, we give you the number 89 that after inverting it and adding it, adding again with the inverted result and repeating this process 24 times, you get a capicúa. Can you tell which capicúa is?

a.- 1883200023881

b.- 8183200023818

c.- 8812300032188

d.- 8813200023188

e.- 88131100113188

References

1. BBC World. 02-02-2020: why is this unusual date called "palindrome day" and what is its correct name. Recovered from: bbc.com/mundo/noticias.
2. López Lasala JL The corner of science. Capicúa hobbies. Recovered from: rincondelaciencia.educa.madrid.org
3. Mathmax. Mathematical hobbies. Recovered from: easymathforallblog.wordpress.com/
4. Mathematics and Poetry. Games and entertainment. Recovered from: mathematics and poetry.com.es
5. Wikipedia. Capicúa. Recovered from: es.wikipedia.com
6. Wikipedia. Palindromic number. Recovered from: en.wikipedia.com