Every route across a five by five grid is ten steps long with exactly five going east, so counting routes is choosing which five of the ten slots are east. The reflex 1024 is the exact number of free ten-step walks, and only 252 of them arrive. Forbid the route to rise above the diagonal and the count collapses to the Catalan number 42.
Four over fifty-two squared is exactly right for the question where the first card goes back, which is what makes it hard to catch. Removing a king shrinks the numerator proportionally more than the denominator, and the counting route through 1326 two-card hands confirms one in 221 without mentioning order at all.
La respuesta refleja ronda las 180, la mitad del calendario. El umbral real es 23, porque una coincidencia necesita una pareja y 23 personas cargan con 253. El mismo razonamiento sitúa la pregunta « ¿alguien comparte MI cumpleaños? » en 253 personas, once veces más gente, y el umbral en raíz cuadrada que hay detrás de ambas explica que un identificador aleatorio de 64 bits se repita tras cinco mil millones de extracciones y no tras dieciocho trillones.