There is a strange playing card phenomenon, with no particular name, that I found in an elusive magic trick book as a child. Some call it the two-card correlation phenomenon, while others call it the Birthday Paradox for playing cards.
I will illustrate below.
Here is a sample of a shuffled deck:
Call out two values, such as 6 and 7. I have personally found that about half of the time, those two cards will be directly next to each other at least once as you spread through the deck:
A bit more than half of the time, there will be exactly one card wedged in between:
Weird, right? The phenomenon pulls from elements of multiple mathematical principles:
Of course, this directly relates to the Birthday Paradox. This paradox proposes that there is a good chance that, in a group of at least 23 people, two of them will share a birthdate. 1
Gilbreath's First Principle states that after you have arranged the cards to alternate in color, cut it in half so that one pile has a black card at the bottom and the other has a red card at the bottom, replaced the cut, and given it a good riffle shuffle, when you alternately deal out the cards into two piles, they will still alternate between red and black. We learn from Gilbreath's First Principle that there is a great mathematical probability of any two given cards being next to one another, even if not specifically red and black in color. 23
The Poisson clumping principle contends that within randomness, events seem to appear in clusters by chance. 4
The Law of Truly Large Numbers concludes that with a large enough number of opportunities, coincidences, such as clumping, appear more common than you would think. 5
Now, go show your friends this mathematical magic trick and watch their reactions.
