Las matemáticas son una ciencia omnipresente. Podemos encontrarlas en cualquier actividad cotidiana, en el mundo que nos rodea y hasta en la naturaleza. Prácticamente todo puede traducirse en números y en esta infinidad de cifras y variables siempre hay lugar para las anomalías y las singularidades. Es así, por casualidad y casi sin querer, que los grandes matemáticos de la historia se toparon con algunos de los fenómenos más curiosos de esta ciencia. Entre ellos, Dattatreya Ramchandra Kaprekar descubrió en el año 1949 un número misterioso, el 6174. Y lo bautizó como la Constante de Kaprekar.
Kaprekar presentó su hallazgo en la Conferencia Matemática de Madrás del mismo año, y desde entonces no dejó de intrigar a los más entusiastas de la materia. Aunque parezca un número trivial y sin ningún significado, el 6174 se convierte en una constante al realizar una serie de operaciones sencillas con números de cuatro cifras. Un hecho que hasta el momento no ha derivado en ningún teorema o proposición dentro de la teoría de números, pero en cualquier caso no deja de ser sorprendente.
Cómo llegar al número 6174, la Constante de Kaprekar
Para comprender la razón de ser de esta constante solamente tenemos que hacer un ejercicio simple:
- En primer lugar, elegimos cualquier número de 4 dígitos con al menos dos de ellos diferentes, pudiendo incluir el 0. Por ejemplo 8675.
- El segundo paso es organizar los 4 dígitos del número en orden descendente. En este caso, 8765.
- Del mismo modo, organizar los dígitos en orden ascendente, 5678.
- Una vez tengamos los dos números, restaremos el más pequeño del más grande. Es decir, 8765 – 5678. Y repetiremos de nuevo el mismo ejercicio. Por lo que:
- 8765 – 5678 = 3087, reorganizamos los dígitos del resultado en orden descendente (8730) y ascendente (0378) y los restamos.
- 8730 – 0378 = 8352, reorganizamos los dígitos de nuevo y los volvemos a restar.
- 8532 – 2358 = 6174
- Llegados a este punto, si repetimos la operación el resultado siempre será 6174, la Constante de Kaprekar.
- 7641 – 1467 = 6174
Si intentamos el ejercicio con otro número diferente, el resultado será el mismo. No importa con qué número comencemos, siempre se repetirá la constante.
Cuando Kaprekar verificó esta «ley» numérica, estableció que el máximo número de pasos necesarios para llegar a 6174 es 7. Y que si tras hacer la operación siete veces no llegas a la constante, has cometido algún error en el cálculo y deberás realizarlo de nuevo.
Este fenómeno no solamente ocurre con números de 4 dígitos con al menos dos de ellos diferentes. Tras descubrir esta constante se ha tratado de investigar el mismo suceso con al menos 2 dígitos y hasta 10. Y se ha repetido con números de 3 cifras, llegando a la constante 495.
Quién fue Dattatreya Ramchandra Kaprekar
A ojos de cualquiera pudiera parecer que este tipo de fenómenos matemáticos no son más que coincidencias entre números. Cosas que no dan explicación a nada y no sirven más que para entretenerse de vez en cuando. Y quizá sea así o quizá no. Mientras algunas casualidades no significan nada, otras le dan sentido a la naturaleza, al crecimiento de las plantas, a la reproducción de los mamíferos y a la proporción natural en que se basa la atracción física entre seres de la misma especie. Como es el caso de la serie de Fibonacci.
Dattatreya Ramchandra Kaprekar fue un matemático nacido en Dahanu, India, a principios del siglo XX. A lo largo de su carrera descubrió muchas propiedades interesantes, además de la Constante Kaprekar, relacionadas con la teoría de números. Incluso se llegó a relacionar su nombre con algunos conceptos «diabólicos» en esta teoría, lo que no deja de ser una muestra de su concienzuda investigación en las matemáticas recreativas.
Otro de sus descubrimientos fue el número Kaprekar, una cifra que elevada al cuadrado y al sumar las dos partes de su resultado, resuelve en el número original. Por ejemplo el 703:
- 7032 = 494,209
- 494 + 209 = 703
Otros números que cumplen esta ley son: 9, 45, 55, 297…
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