Puzzle: Crystal Ball Connection Patterns

K4 matchings

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.

TheWizardBySeanMcGrath-small

  • Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?
    T(0) = ?
    T(1) = ?
    T(2) = ?
    T(3) = ?
    T(4) = 10 connection patterns (as above)
    T(5) = ?
    T(6) = ?
    and so on.

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.


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Calling All Math Teacher Bloggers and Homeschoolers: Carnival Time!

by Bob Jagendorf via flickr[Image by Bob Jagendorf (CC BY-NC 2.0) via Flickr.]

The monthly Math Teachers at Play (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

Don’t procrastinate: The deadline for entries is this Friday, May 22. The carnival will be posted next week at ZenoMath.

Would You Like to Host the Carnival?

Thank you so much to the volunteer bloggers who have stepped up to carry this MTaP math education blog carnival through the summer! I would never be able to keep the carnival going on my own.

If you’d like to join in the fun, we have openings for the 2015-2016 school year. Read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:


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New Fantasy Novel by Homeschooled Teen Author

Hunted600After months of editing, formatting, proofreading, sweat, and tears:

Teresa Gaskins’s new ebook Hunted: The Riddled Stone ~ Book Two is available now at Amazon. The paperback should follow within the next couple days, and the other online retailers will come along whenever their automated systems get caught up.

You can download the first five chapters here:

To celebrate the release of Hunted, the ebook version of Banished‌—‌the first book in the Riddled Stone series‌—‌will be on sale for 99 cents for the next few weeks.


Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Math Game: Fan Tan (Sevens)

Feature photo above by Morgan (meddygarnet) via Flicker (CC BY 2.0).

Math Concepts: sorting by attribute (card suits), counting up, counting down, standard rank of playing cards (aces low).
Players: two or more, best with four to six.
Equipment: one complete deck of cards (including face cards), or a double deck for more than six players. Provide a card holder for young children.

How to Play

Deal out all the cards, even if some players get more than others. The player to the dealer’s left begins by playing a seven of any suit. If that player does not have a seven, then the play passes left to the first player who does.

After that, on your turn you may lay down another seven or play on the cards that are already down. If you cannot play, say, “Pass.”

Once a seven is played in any suit, the six and the eight of that suit may be played on either side of it, forming the fan. Then the five through ace can go on the six in counting-down order, and the nine through king can go on the eight, counting up. You can arrange these cards to overlap each other so the cards below are visible, or you can square up the stacks so only the top card is seen.

A Fan Tan game in progress.
A Fan Tan game in progress.

Players do not need to wait for both the six and eight of a suit to be played before they begin building the fan up or down.
The first player to run out of cards wins the game.

If you want to keep score, count the cards remaining in your hand after one player goes out. After everyone has had a turn as dealer, whoever has the lowest total score is the champion.

Variations

In some traditions, play always begins with the seven of diamonds, so whoever has that card goes first.

Domino Tan

The player to the dealer’s left may lead any card, and then all the suits must start with that number (instead of with seven) and build up and down from there.

Fan Tan Trumps

When the dealer gets to the end of the deck and there aren’t enough cards to give every player one more, the last few cards are turned face up and may be played by anyone as needed. The suit of the last card becomes the trump suit, and cards of that suit may be played on any of the fans, with the card they replace going on the trumps fan. In this case, the cards must be laid out in overlapping rows, not stacked up, so everyone can see where the trumps have gone.

For instance, if spades are trump, then a nine of spades could be played on the eight of hearts, which would leave the nine of hearts without a home—so it has to go on the spades fan.

Exceptions: The seven of the trump suit starts its own fan, like any other seven, and the last card dealt (the one that named the trump suit) must also be played to the trumps fan when its turn comes.

Crazy Tan

Deal only seven cards to each player, and set the rest of the deck out as a draw pile. The first player who cannot play must draw one, which he may play if possible. If not, and the next player also cannot play, she must draw two. If neither of those cards will play, and the next player has nothing to play, he must draw three, and so on, each player drawing one more card than the last person. When one of the players is finally able to lay down a card, this resets the draw count back to zero.

In Crazy Tan, players are allowed to lay down a run (playing several cards in a row of the same suit on a single turn). Or they may play parallel cards (cards of the same rank in different suits, all played in the same turn). Or a player may even lay down parallel runs, if the cards happen to work out that way.

History

Fan Tan may also be called Crazy Sevens. Like any folk game, it is played by a variety of rules around the world. If you search for it on the Internet, you may run into an unrelated Chinese gambling game called Fan Tan, which is similar to Roulette.


Counting-Games

This post is an excerpt from my book Counting & Number Bonds: Math Games for Early Learners, available now in bookstores all over the Internet.


Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Playful Math Snacks for May 2015

Tabletop Academy Press UpdatesThe May “Let’s Play Math” newsletter went out Monday morning to everyone who signed up for Tabletop Academy Press math updates. This month’s issue focuses on math games, from Rosie’s Princess in the Dungeon to Jim Pai’s Trig & Logarithm War. What fun!

If you’re not on the mailing list, you can still join in the play:

And remember: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

A Preview

Math Snack: Two Math Games
Playful, no-preparation math activities for all ages

Math games pump up mental muscle, reduce the fear of failure, and develop a positive attitude toward mathematics. Through playful interaction, games strengthen a child’s intuitive understanding of numbers and build problem-solving strategies. Mastering a math game can be hard work, but kids do it willingly because it is fun.

The easiest no-prep strategy game for all ages is the finger game Chopsticks. You don’t need any equipment, so it’s a great game to keep in mind for when you and your kids are stuck in a line or waiting room. Math concepts: counting up to five, thinking ahead.

And one of my all-time favorite games is…


Feature photo at top of post by marcello (CC BY 2.0) via Flickr.


Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Math Games with Factors, Multiples, and Prime Numbers

Students can explore prime and non-prime numbers with two free favorite classroom games: The Factor Game (pdf lesson download) or Tax Collector. For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or try the following game by retired Canadian math professor Jerry Ameis:

Factor Finding Game

FactorFindingGame

Math Concepts: multiples, factors, composite numbers, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal, vertical, or diagonal) wins. Or for a harder challenge, try for four in a row.


Feature photo at top of post by Jimmie via flickr (CC BY 2.0). This game was featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #79. Hat tip: Jimmie Lanley.


Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Math Teachers at Play #85

[Feature photo by Tomruen via Wikimedia Commons.]

MTaP-85

Do you enjoy math? I hope so! If not, browsing the articles linked in this post just may change your mind.

Welcome to the 85th edition of the Math Teachers At Play math education blog carnival‌—‌a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.

By tradition, we start the carnival with a short puzzle or activity. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Let the mathematical fun begin!

Continue reading Math Teachers at Play #85

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