“Math Teachers at Play is a monthly blog carnival that compiles posts from several blogs under the theme of teaching math in unique and innovative ways. We’re thrilled to partner with them this edition and couldn’t help ourselves from including some interesting trivia and riddles centered on our edition number – Happy Number 86!
Happy Numbers are numbers whose digits are such that – when squared and added iteratively, the number 1 is reached. For example,
82 + 62 = 100
12 + 02 + 02 = 1
So 86 is a happy number!…”
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.
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.
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.
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.