The Guess Who Conundrum

Every so often, an idea comes out of left field. I woke up with this on my mind--must have been a dream.

Back in the day, my family had a dilapidated copy of the game "Guess Who?" My siblings and I would take turns playing this game of deduction. You essentially narrowed a search for an opponent's person by picking out characteristics of their appearance.

http://www.flickr.com/photos/unloveable/2398625902/

I vividly remember playing with my younger sister one time at a family cottage. She--foolishly--chose a female person for me to identify. Anyone who has played the game before knows that the males far outnumber the females. I had her narrowed down to six or seven possibilities with one question:

"Is your person a male?"

Move the same principles into the math classroom. Numbers have characteristics much like Robert, Frans, and Bob. Some are large, negative, prime, abundant, triangular, square, etc. Would it be possible to play a game of "Guess Who?" with an array of numbers?

Create a five-by-five game board with numbers from 1-100 dispersed on it. Sample boards are available for download. Both players must have the same numbers on their boards for the game to work. They can be arranged differently to avoid peeking. It would be wise to choose a variety of numbers. Some even, some odd, some prime, some multiples of 3, etc. Try to avoid one dominating characteristic. (Like my sister's gender choice).

Have each player choose a number, and play the game. I would place restrictions on what they can and cannot ask:

1. Must be a yes-or-no question
2. Can't pertain to a physical characteristic of the symbol. (Does your number have curves?)
3. Can't ask if the number contains specific digits (Does your number have an 8?)
4. Can't ask over-under questions. (Is your number over 12?)

Encourage students to ask if numbers are prime, square, cubic, multiples of specific integers, triangular, odd, even, etc. Be sure they are dealing with the characteristics of the numbers.

After a round or two--depending on the group speed--get down to the real task. Have students pair up with their opponent and answer the following question:

Which number is the best to choose and why?

Students will begin brainstorming the most likely questions, and which characteristics pertain to which numbers. They will be practicing prime factoring within the framework of a larger task.

Students are essentially looking for the number that shares the most characteristics with the other numbers. Any unique characteristics run the risk of being singled out with a single question. This categorization of numbers fits very well in the Grade 10 Pre-calculus factors and products curriculum in my province. These topics are fairly universal across school mathematics.

@natbanting are we only looking at whole numbers?
— David Crowell (@cageyprincipal) August 18, 2012

Great question. I bet characteristic games can be applied to the rational numbers as well. I can see a less than--greater than approach working well for ordering of fractions. 

It might be interesting to have students play in pods of three. Each person has to deduce the other two's numbers by finding common characteristics. Every question is posed to the pair of opponents and each must give a yes-or-no response. Maybe they are both even, but only one is a multiple of 6? Maybe one is even and the other is odd but they are both perfect squares? The game ends when a player correctly guesses both opponent's numbers with a single guess. This throws an interesting wrench into the deduction gears. It also provides a nice extension for those students who are ready to move onward while others need more time with basic factoring. 

This is a great way of throwing mathematical deduction and problem solving alongside factoring. Students should see numbers like they see eccentric faces--as culminations of a series of defining characteristics. 

NatBanting