Modular Arithmetic: Family Vacation Trick (Interview)
There exists 4 families: the Diamonds, the Hearts, the Clubs, and the Clovers. They are all staying at the Heart Ace Hotel. But what happens when the families start to fight? They all just want to be with their friends. How can they break it up to make the 4 room arrangement work? Follow the steps below!
How did the familes get resorted? See explanations below!
The image below shows the backside of the cards after step 3:
The Families get Together. One can see that the order in their "stack" has
has remained. Note that each suit corresponds to a different mod.
Namely, Kings are mod 1, Queens are mod 2, Jacks are mod 3, and 10s
are mod 4.
During step 4: The Families Break (3 times), cards are broken
at a given point and moved to the back of the deck. Notice that
the card's position changes, but the mod of its suit does not. The King of Hearts was in position 1, and after the switch
it is in position 12. However, before and after the switch, all
Kings exist in the same mod!
This is because of modular arithmetic! Dealing with 16 cards, in a mod 4 system (the suits repeat every 4 cards), a card's mod depends more on its order than its "number" in line! In the first picture, the Queens are in positions 2, 6, 10, 14. After they are in positions 1, 5, 9, 13. Their position changes, but their mods are always the same, as shown below.
Thus, when the "families" are distributed in a counterclockwise direction, the mods always line up, or each family member gets to room with their friends!
