Tova Brown continues to examine Hilbert’s Hotel Paradox, pondering infinite sets of infinite sets.
As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full. Your hotel becomes very popular among infinite sports teams, as a result.
Recruitment season presents a challenge, however, when many infinite teams arrive at once. How many infinite teams can stay in a single infinite hotel?
Tova Brown explores the second part of Hilbert’s Hotel Paradox. What’s infinity plus infinity?
Running an infinite hotel has its perks. Even when the rooms are full you can always find space for new guests, so you proudly welcome everyone who appears at your door.
When two guests arrive at once, you make room. When ten guests arrive, you accommodate them easily. When a crowd of hundreds appears, you welcome them all.
Is there no limit to your hospitality?
Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…
Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.
But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.
[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]
Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!
For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.
Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
High school math teacher Chris Rime has done it again. Check out his November 2015 printable math calendars for Algebra 1 (in English or Spanish), Algebra 2, and Geometry students. Enjoy!
At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).
In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.
As a puzzle:
Cut the calendar squares apart and trim off the dates. Then challenge your students to arrange them in ascending (or descending) order.
Make up problems to fill a new calendar for next month.
And if you do, please share!
Check out this new puzzle book for upper-level high school students & adults:
Thomas Povey is a Professor of Engineering Science at the University of Oxford, where he researches jet-engine and rocket technology. In his new book Professor Povey’s Perplexing Problems, he shares his favorite idiosyncratic stumpers from pre-university maths and physics.
These problems “should test your ability to grapple with the unfamiliar,” Povey writes. “You will learn to tease new problems apart, and apply things you already know in ways you had never considered. You have all the tools you need, but you should see what amazing things you can do with them.”
Alex Bellos shared one of Professor Povey’s puzzles in The Guardian. Can you figure it out?
The book starts off with geometry, but most of the chapters focus on various topics from physics. Some of the puzzles are accessible through applied common sense, but for many of them, it helps to have taken an algebra-based (high school level) physics course.
Kitten is just finishing up her physics textbook, and she still has one more year of homeschooling. I’m hoping to work several of these puzzles into our schedule this year. It should be great fun!
If like me you’re a bit rusty on your physics, don’t worry. Each answer is thoroughly explained—in fact, it takes a bit of discipline to close the book and try your hand at each problem before reading on. I wish they’d put the solutions in the back rather than in the main text, to make it easier to browse the problems without reading spoilers.
Speaking of which, here’s the answer to the video puzzle above…
The August “Let’s Play Math” newsletter went out last week to everyone who signed up for Tabletop Academy Press math updates. This month’s issue focuses on logic puzzles for all ages, including a newly-discovered deleted scene from Harry Potter and the Sorcerer’s Stone. What fun!
If you missed this month’s edition, no worries—here are some great puzzles from the Let’s Play Math blog archive:
If you missed this month’s edition, no worries — there will be more playful math snacks coming soon. Click the link below to sign up today!
And remember: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.
One of the most persistent math myths in popular culture is the idea that mathematics is primarily about getting right answers.
The truth is, the answer doesn’t matter that much in math. What really matters is how you explain that answer. An answer is “right” if the explanation makes sense.
And if you don’t give an explanation, then you really aren’t doing mathematics at all.
Here is a short sequence of numbers. Can you figure out the rule and fill in the next three blanks?
2, 3, 5, 7, ___, ___, ___, …
Remember, what’s important is not which numbers you pick, but rather how you explain your answer.
Perhaps the sequence is the prime numbers?
2, 3, 5, 7, 11, 13, 17, …
The prime numbers make a wonderful sequence, though it isn’t the one I was thinking of.
Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits
Prerequisite: able to identify fractions as part of a whole
This is how I tell the story:
Continue reading Infinite Cake: Don Cohen’s Infinite Series for Kids
In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.
Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.
On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.
Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.
For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.
Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.
Feature photo at top of post by Christian Schnettelker (web designer) and wizard photo by Sean McGrath via Flickr (CC BY 2.0). This puzzle was originally featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #76.
Denise Gaskins' Let's Play Math 
