Introduction
The concept of Mathematics started a long time ago. The origins of mathematical thought lie in the concepts of number, magnitude an form. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. Predynastic Egyptians of the 5th millenium BC pictorially represented geometric designs and it has been claimed that megalithic monuments in England and Schotland, dating from 3rd millenium BC, incorporate geometric ideas such as circles, ellipses and Pythagorean triples in their design,...
Mathematics as we all know now, has been developed by different cultures and civilitations, and like we have now, they struggled to find solutions to the problems they were finding.
Task
We are going to learn and solve some problems based on games from different cultures, and to do that, you are going to read about topics that you will apply to solve these problems. Also, each problem will describe the game and the background.
Process
Evaluation
Problem 1. (Israel)
Background.
One of the best known symbols of Chanukah is the Dreidel. A dreidel is a four sided top with a Hebrew letter on each side. They stand for "Nes Gadol Hayah Sham", which means "A great miracle happened there." In Israel the dreidel has pey instead of shin, for "Nes Gadol Hayah Po," which means "A great miracle happened here." The miracle is that the small flask of oil burned the menorah lamp for eight days. Dreidel is a popular game played during the Holiday.
Description. The dreidel is a four-sided top, with the following on the four faces.
(N) or nun stands for nicht or nothing. If the dreidel lands on nun, you do nothing.
(G) or gimmel stands for ganz or all. If the dreidel lands on gimmel, take everything in the middle.
(H) or hay stands for halb or half. If the dreidel lands on hay, take half of what's in the middle plus one if there's an odd number of objects.
(SH) or shin stands for shtel or put in. If the dreidel lands on shin, put two objects into the middle.
- If the dreidel is fair, what outcomes would you expect if you spin it 100 times?
- Are the outcomes of spinning the dreidel independent from each other? why or why not?
Problem 2 (Mexican)
Background. Toma todo or pirinola is a game of chance from Mexico. Children as well as adults play Toma todo with a six-sided spinner (pirinola) or top. Adults often play the game using money.
Description. Toma todo is a game of chance in which each of the six possible outcomes on the pirinola is equally likely to occur. Each player seeks to acquire all the beans.
Rules.
-
To begin the game, each player places one bean in a center pile. Turns are taken. A turn consists of spinning the pirinola and following the directions on the side that is facing up after the spin.
-
The following are directions to be written on the pirinola:
Pon 1 (Put 1).
Pon 2 (Put 2).
Take all (Toma todo).
Take 1 (Toma 1).
Take 2 (Toma 2).
Each player puts 1 in the pile (Todos ponen).
• If toma todo turns up, then in order to continue the game, each player must place one bean in a center pile again. Play continues until one player has all the beans or candies.
- It appears that the pirinola is fair and each of the six sides is equally likely. What is the probability of each side (outcome) if this is true?
- Are the outcomes of spinning the toma todo independent? Why or why not?
- What is the probability that two player in a row get the same outcome when they spin the pirinola?
Problem 3 (Native American)
Background. This is another simple game of chance. Native American peoples played a number of stick games with varied rules.
Description. Four sticks are tossed and scored depending on whether they fall on the face or reverse side.
Rules.
• Place the toothpicks or beans in the center of the playing area. Decide in advance how many rounds to play. Take turns. Hold the sticks in one hand, and let them fall to the ground or the table.
• Scoring:
o All four up, 5 points
o Three up and one down, 2 points
o Two up and two down, 1 point
o One up and three down, 2 points
o All four down, 5 points
• Count the points earned and take that number of toothpicks or beans from the pile, and place them next to you. The player with the greatest number of toothpicks or beans is the winner.
-
Is this a fair way to score the game? Four sticks can fall in sixteen different ways. Make a table of all of the ways the sticks can fall.
-
Some of the sixteen ways in question one have the same outcome, since order does not matter. How many different outcomes are there? What is the probability of each outcome?
Problem 4 (Navajo)
Background. Ashbii (ash been) is a game played by Navajo women and children. The game was observed in the town of Chin Lee Arizona on the reservation there. The Navajo women sometimes played this game while sitting under a buffalo hide which had been staked up for drying.
Description. The game is played with three painted sticks and a basket. The sticks act as dice and are tossed.
Rules.
All players sit on the floor. The sticks are tossed upward, ideally against a blanket stretched overhead, and the sticks that land in the basket are scored. Only the highest scoring combination that occurs on any one toss will count. Players alternate throwing the sticks. The winner is that player who first scores 25 points.
Scoring:
1 point for all black or half black.
2 points for all red or half red.
3 points for crossing (all black and all red, all black and half red, all red and half black, half red and half black).
5 points for crossing all red and half red.
1. Is this a fair way to score the game? How many different ways can the sticks fall. Make a table of all of the ways the sticks can fall. Don’t forget those which do not earn points.
2. How many different outcomes are there? What is the probability of each outcome?
Problem 5 (Chinese)
Background. NIM is an Ancient game that originated in China.
Description. NIM is a simple game with few rules. Beginning with 11 beans, two players alternate turns of taking away 1 or 2 beans. The player to remove the last object loses.
Rules.
-
The two players alternate turns.
-
There is a pile of 11 beans between the two players.
-
Each player must remove either 1 or 2 beans in turn. A player may not skip a
turn.
-
The player to remove the last object loses.
1. What strategies could you use to win this game?
2. Is it better to be the first or second player?
3. What is the probability of winning if you are the first player and you know the rules? When you are done with all 5 problems, check your answers with your peers and, in case your answers differ, discuss why you are right or wrong, using Probability
Conclusion
Congratulations! You learned probability using games and problems from different cultures! I am sure you knew some of the games. If not, now you can play them!
In case you want to make them, here are the necessary materials for each game:
For the game on Problem 1.
Dreidels can be purchased at holiday stores and toy stores. Inexpensive plastic or wooden ones are available. You might also be interested in seeing quite expensive silver or crystal dreidels. Players use pennies, nuts, raisins, or chocolate coins (gelt) as tokens or chips.
For the game on Problem 2.
Several beans or wrapped hard candies
1 six-sided pirinola (available in Mexican gift shops)
For the game on Problem 3.
Each pair will need:
4 popsicle sticks or tongue depressors decorated on one side with markers Toothpicks or beans to keep score.
For the game on Problem 4.
Each game will require three sticks.
-
The first is called the “tsi’i (zeen) head.” It is painted completely black on one
side and half black on the other.
-
The second stick is called “nezhi’ (nezshi), and is painted half red on one side and half black on the other side.
-
The third stick is called “tqelli” (zelli) and is painted all red on one side and all black one the other.
-
Large basket or paper plate to catch sticks.
For the game on Problem 5.
A pile of beans or other similar objects.
Credits
- Khan Academy
- Mathematics activities from different cultures, 2004