The world is a confusing place and we humans struggle to parse all the information thrown at us on a daily basis. The one thing we can rely on through all our struggles is logic and problem-solving skills, right? Not so fast! In a recent Berkeley Connect Math discussion, Berkeley Connect Mentor and math PhD candidate Kubrat Danailov explored the crazy world of paradoxes: times when logic fails and problems become irresolvable.

During this discussion, Danailov introduced students to a series of mathematical paradoxes, each one a

complicated brain-teaser waiting to be solved, including the Achilles and the Tortoise paradox, Hilbert’s Hotel, Russel’s Paradox, and the Social Media paradox. How do these paradoxes work exactly?

Imagine a hotel with infinitely many rooms and in each room is a guest. Logically, you might think that this hotel is full, but mathematician David Hilbert would beg to differ. In the aptly named “Hilbert’s Hotel” paradox, a guest being in every room doesn’t mean new guests can’t fit, and as Danailov explained, the hotel can fit an infinite number of new guests. How is this possible? Students pondered this question in groups, attempting to identify the resolution to such a paradox. The answer is simpler than you might think!

Because there are infinite rooms, guests can be accommodated through the simple shifting of guests from room to room. If a new guest arrives at the hotel, each guest currently in the hotel will be shifted one room down to accommodate the new arrival. Guest #1 in room #1 moves to room #2 while guest #2, formerly in room #2 moves to room #3 and so on, to infinity. What if, however, an *infinite *number of guests were to arrive at the hotel, all wanting a room? Instead of shifting every guest one room over, each guest now moves to a room twice as high a number as they are. In other words, guest #1 will move to room #2, guest #2 will move to room #4, guest #3 will move to room #6, and so on to infinity! This leaves all odd-numbered rooms accessible to the infinite number of new guests, resolving the paradox.

Students discussed the concept of infinity in-depth amongst themselves, as it related to solving Hilbert’s Hotel. This paradox helps to illustrate that infinity + infinity = infinity, as does infinity * infinity + infinity = infinity, making clear just how large a concept infinity is.

Also discussed during this section was the Social Media paradox, a social phenomenon that people experience when they realize that most of the people on their friend’s list, on average, have more friends than they do, which contradicts people’s belief that they have more friends than their friends do. These perceptions conflict and create an irreconcilable logical breach in our thinking patterns.

Math is often an area we look towards for absolute answers, but the six unique paradoxes (only a small sampling) that Danailov shared with students demonstrate that even logic is not always crystal clear. Paradoxes are a great topic to get the brain engaged, and each paradox on this list is worth looking into, some more difficult than others. You don’t need a math background to get involved and see if you can solve these challenges!

*posted by Dylan McIlvenna-Davis, **Berkeley Connect Communications Assistant (Class of ’20)*