Two-color Chip Game
Thanks to Nicholas Branca, San Diego State University
Decide which partner will be player A and which will be player B.
Game 1
- Begin with two chips of one color and one of the other color in the bag.
- Shake the bag.
- Each player pick one chip from the bag (without looking).
- Person A wins if the chips are the same color; person B wins if the chips are different colors
Is this a fair game?
Game 2:
Replay game 1 with three of one color chip and one of the other color.
Is this a fair game?
For what values of n and m (where there are n chips of one color and m chips of the other color in the bag) will this game be fair? Find as many pairs of numbers n and m as you can. When you think you have found a pair which produces a fair game, justify your claim. Try to generalize and justify your results.
Writing up your work: Follow the guidelines below.
1. Restate the Problem clearly enough so that someone who is unfamiliar with the problem could understand what it is you are asked to do.
2. You will probably make many notes in route to solving the problem. Based on these notes, describe what you did (process) in attempting to solve this problem. Include descriptions of how you got started, what approaches you tried, where you got stuck, and what drawings you used. Include things that didn’t work out or that seemed like a waste of time. Do this portion of the write-up even if you did not solve the problem.
3. State your solution or solutions as clearly as you can. Explain how you know that your solution is the best one possible or as complete as possible. Make it convincing so that even someone who initially disagrees with your ‘answer’ will have reason to believe you are correct. Merely stating your solution does not meet the requirements of this section.
4. Invent some extensions or variations to this problem. Write down some related problems that may be easier, more difficult, or the same level of difficulty as the original problem. You do not have to solve these extensions.
5. Define and Evaluate the mathematical content of the problem. What mathematical topics or processes were required for this investigation? Evaluate the value of this investigation for your development in mathematics and communication of mathematics.
Discussion of the “Fair Games” problem, based on a conjecture that grows out of an analysis of several games:
If w represents the number of white chips in the game and b represents the number of blue chips in the game, then the fairness function for the game: F(w,b) is FAIR whenever w and b are consecutive triangle numbers (consecutive terms of the sequence 1, 3, 6, 10, 15, 21, …).
What someone does to get to this point is to count out the number of ways for person A to win the game (same color) and the number of ways for person B to win the game (different color). If these two counts are the same, then the game is fair!
- Counting the number of ways that person A can win:
Person A wins by having either two white chips or two blue chips drawn. The number of ways to draw either two white or two blue chips follows the same pattern – each is a count of the number of ways to select a pair of objects from a set of objects. Note the pattern of such counts:
All possible pairs
Number of objectsExample set of objects(order does not matter)No. of pairs
2{a1,a2}a1a2 1
3{a1,a2,a3}a1a23
a1a3
a2a3
4{a1,a2,a3,a4}a1a26
a1a3
a1a4
a2a3
a2a4
a3a4
5{a1,a2,a3,a4,a5}a1a2 10
a1a3
a1a4
a1a5
a2a3
a2a4
a2a5
a3a4
a3a5
a4a5
Hmmm… This is a familiar pattern… let’s generalize:
Number of objectsNumber of pairs of objects(order does not matter)
w white chipsw(w-1)/2
b blue chipsb(b-1)/2
Hence, the number of ways for person A to win our game is given by the expression
.
- Counting the number of ways that person B can win:
For person B to win, there must be one chip of each color. Since there are w choices for the white color and b choices for the blue color, the total number of combinations is wb.
Hence, the number of ways for person B to win our game is given by the expression
Wb.
The game is fair if and only if
= wb.
Let’s manipulate this equation a bit:
Our analysis shows that for our game to be fair, w and b must be two numbers whose sum is the square of their difference. Now let’s connect the property, “two numbers whose sum is the square of their difference” to the triangle numbers.
Note that any two consecutive triangle numbers can be written in the form:
for some natural number n.
Their sum is: .
The square of their difference is .
Hence, for any two consecutive triangle numbers, their sum is the square of their difference.
Wow!