by Dave Youngs
Magic cards based on the binary number system have been a part of recreational mathematics for years. They are used to play a game in which "mind readers" (those who know the cards' secret) shows the cards to players (those who don't know the secret) and ask them to think of a number in a certain range (usually from 1 to 31). The mind readers say that the number can be guessed correctly if they are told on which of the cards the chosen number appears. When the mind readers are given this information, they immediately tell the players the correct number. The mind readers do this by adding the first number on each card chosen. This works because each of these numbers represents one of the binary number system's place value groupings ( 16's, 8's, 4's, 2's, and 1's) used to make the players' number. For example, 23 appears on the cards with 16, 4, 2, and 1 in the top left corners, these add to 23.
Teddy Bear Magic Cards is an activity based on a simple version of the above game. It uses four cards, each with eight numbers that range from 1 to 15. Before doing this activity with students, make a set of magic cards large enough to be seen by all when the cards are displayed at the front of the room. Use the numbers and colors indicated on the student page.
Introduce the activity by showing students the magic cards and telling them that you can discover the numbers (from 1-15) they are thinking of if they tell you what cards the numbers are on. Ask if any of the students have played this game before. If some already know the secret, ask them not to give it away and tell them they can help you guess other students' numbers. Play several games with the class before challenging students to discover the secret of the magic cards.
At this point, you need to decide how to proceed with the activity. You may want to have students work independently on the student pages or you may choose to have students work in groups. Another decision you need to make is what to do when students discover the secret, or what to do if no one discovers it. If one student, or group of students, tells the rest of the class the secret, it robs others of the opportunity to discover the secret on their own. In some cases, the frustration level might indicate that it would be appropriate to share the secret and move on to the next activity. Think these things through before doing the activity with your class.
The first student page provides each student with a copy of the magic cards that can be taken home and used to play the game with family members. The second page is a chart which is designed to help students see some patterns that emerge from the magic cards. To complete the chart, students fill in the boxes that correspond to the numbers on each card, using the colors indicated. For example, Card A has the numbers 8-15 on it, so the boxes under that card corresponding to 8-15 are colored in red. Likewise, Card B contains the numbers 4-7 and 12-15, so the boxes corresponding to these numbers would be colored in green. The same procedure is followed for Cards C and D.
The completed chart can be used to guess other people's numbers. For example, if the number chosen is on Card A, a look at the chart will eliminate half the numbers (0-7), since these do not use an eight and are not colored in. Finding out if the number is on Card B will narrow the field by half once more. Likewise, Card C will cut the remaining possibilities in half leaving only two choices. Knowing if the number is on Card D will then give the exact number.
Using this chart is really a better way to discover someone's number than just adding the first number on each card chosen (which is the secret given in most books that have magic cards). Hopefully, students will discover this.
I hope you enjoy this activity.