A capicúa number It is the one that is read the same from left to right, as from right to left. The Royal Spanish Academy reserves the floor palindrome when it comes to 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's a palindromic phrase.
The word capicúa originates from Catalan: cap i cua what does it mean head Y tail, and applies exclusively to numerical figures that are read the same in both directions.
These figures are also called palindromic 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 this type of number is called "palindromic numbers ".
Article index
1.- Any one-digit number is considered a capicúa: 0 is a capicúa, 5 is a 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 central equals itself.
3.- If the number of digits of 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.
Let's consider 73137, in this number 1 is the central digit and center of symmetry, 3 is at distance one from the central digit and 7 is at distance two from the central digit in both directions.
4.- If the number of digits is pair, 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.
8.- A sequence with a finite number of terms is capicua if the equidistant terms are equal: a0, to1,… , toi,… , ton-1, ton is capicúa if ai = an-1.
9.- When the sum of the n elements of a geometric progression, which begins at 1 and has the 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.
On Sunday, February 2, 2020, he was baptized as the "Palindrome day", during that week the label #PalindromeDay it was a trend in the networks.
The reason is that this date is capicúa in the format day month Year, in the format month day Year, or in the format year month day:
02/02/2020
2020/02/02
The date prior to 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 indicated day: 12/12/2121
In the day / month / year format, the capicúa before the #PalindromeDay It was 10 years ago and one day: 02/01/2010, while the next 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..
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.
I have four digits, I am 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
a.- 5, b.- 7, c.- 10, d.- 52, e.- 89
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 have to 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
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