A Math Forum Project

ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Galactic Exchange - posted November 6, 2000

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Introduction

It's the year 2075. You take a vacation to the planet Orange. When you arrive, you notice coins of three different shapes: squares, circles, triangles. Unfortunately, no one speaks your language, and you're hungry. You find a food vending machine, but the prices are worn off! You have to try different combinations of coins to get food, and in addition, you will be able to figure out the relations between the coins.

Applet

Click here to show vending machine

Questions:

1. Buy a package of zoogs. What are the fewest number and types of coins needed to purchase a package of zoogs using exact change?
   
   
2. Which coin is worth the least, and how many of it does it take to equal each of the other coins?
   
   
3. How many of the least valuable coins would it take to buy a package of zoogs?
   
   
4. Buy a package of Glorps. What are the fewest number and types of coins needed to purchase a package of Glorps using exact change? How many of the least valuable coins would it take to buy a package of Glorps?
   
   
5. Another Earthling arrives on the next shuttle and wants to try some Mushniks. Explain how you figured out the relations between the coins.
   
   
6. What are the fewest number and types of coins needed to purchase a package of Mushniks using exact change?

Additional Practice

If you would like to practice this puzzle with different values, go to our practice page. These puzzles are for practice -- you will not submit your solutions to the Math Forum. Enjoy!

Click To Show Practice Window

Comments

Teacher Support Page

There were some problems using the applets early in the week, but they were fixed later and people started submitting solutions. Thanks for your patience!

The main thing we were looking for was a good explanation about how you figured out the relations among the three coins, which was asked in question 5. The other questions were meant to give you an idea of how to solve the problem, and to give us some indication of what kinds of problems you were having.

The main problem people had this week was that they didn't answer all the questions. A second problem was that people weren't entirely clear about how they went about solving it. Look at the highlighted solution below to see a great explanation.

Some people told us they bought the products with an incorrect amount of money, and some people gave us amounts that included change even though we asked for the exact price, i.e., without change.

If you really like solving problems like this, at the bottom of the problem page (also archived) is a practice applet that has 10 different combinations of coin values and product prices.

Highlighted solutions:

I wil present the answer to question 2 first, because it will help
answer all the other questions.

Notation: I will denoted Squares by s, Circles by c and Triangles
by t.


2.Which coin is worth the least and how many does it take to equal
each of the other coins?

Answer:  The squares are worth the least. It takes 3s to equal 1c and
it takes 7s to equal 1t. I found this out by experimenting with the
vending machine, and by finding out how many coins of each type are
needed to buy a package of zoogs. We get the equations:

(1)   8s = a package of zoogs,
(2)   3c - 1s =a package of zoogs
(3)   2t - 2c =a package of zoogs.

I will use equations (1) and (2) to find the relation between 1s and
1c. Because the left hand sides of both equations (1) and (2) equal
one package of zoogs they equal each other, so we can make the
following equations:
3c - 1s = 8s
3c = 9s   (added 1s to both sides)
1c = 3s   (divided both sides by 3).

Now that we know the relationship between 1s and 1c, we can find out
the relationship between 1s and 1t by using the equations (1) and (3).
Again the left hand sides of both equatians equal a package of zoogs,
so we can make the following equation:
2t - 2c = 8s
2t - 6s = 8s  (substituted 2c for 6s)
2t = 14s       (added 6s to both sides)
1t = 7s         ( divided both sides by 2)


3.How many of the least valuable coins would it take to buy a package
of Zoogs?

Answer: As pointed out in the solution for 2, the least valuable coin
is the square, and it takes 8s to buy a package of zoogs.


1.Buy a package of Zoogs. What is the fewest number of coins needed to
purchase a package of Zoogs using exact change? Make sure you list
types of coins as well.

Answer: The smallest number of coins to buy a package of zoogs without
change is 2. Indeed, we now know that a package of zoogs = 8s =
7s + 1s = 1t + 1s.


4.Buy a package of Glorps. What is the fewest number of coins needed
to purchase a package of Glorps using exact change. Include the types
of coins in your answer. How many of the least valuable coins would it
take to buy a package of Glorps?

Answer: The fewest number of coins needed to buy a package of glorps
is 3, namely 1s + 1c + 1t.  Indeed,   by experimenting with the
vending machine, we found out that a package of glorps costs 11s. This
equals 7s + 3s + 1s = 1t + 1c + 1s.


6.What is the fewest number and types of coins needed to purchase a
package of Mushniks using exact change?

Answer: The fewest number of coins needed to buy a package of mushnics
is 4, namely 2t + 2s. Indeed, by experimenting with the vending
machine, we found out that a package of mushniks costs 16s. This
equals 14s + 2s = 2t + 2s.

5.Another Earthling arrives on the next shuttle and wants to try some
Mushniks. Explain how you figured out the relationship between each
coin.

Answer: I would show to the other Earthling my solution to questions 2
and 6 (as presented above), so that he/she knows how to buy mushniks.

20 students received credit this week.

Alex Acosta, age 15 - Westside High School, Houston, TX
Maninder B., age 13 - Caroline Davis Intermediate School, San Jose, CA
Lee Bobbitt, age 15 - Westside High School, Houston, TX
Jasmine Branch, age 17 - Westside High School, Houston, TX
Joey C., age 13 - Caroline Davis Intermediate School, San Jose, CA
Jerson, Jr. Cometa, age 18 - Westside High School, Houston, TX
An D., age 14 - Caroline Davis Intermediate School, San Jose, CA
Hoang D., age 13 - Caroline Davis Intermediate School, San Jose, CA
Elizabeth G., age 12 - Caroline Davis Intermediate School, San Jose, CA
Andrew H., age 12 - Caroline Davis Intermediate School, San Jose, CA
Denise H., age 13 - Caroline Davis Intermediate School, San Jose, CA
Josh H., age 13 - Caroline Davis Intermediate School, San Jose, CA
Shannon M., age 13 - Caroline Davis Intermediate School, San Jose, CA
Tony M., age 13 - Caroline Davis Intermediate School, San Jose, CA
Donielle Miller, age 17 - Westside High School, Houston, TX
Long N., age 13 - Caroline Davis Intermediate School, San Jose, CA
Mihai N., age 11 - Mary Johnston Public School, Waterloo, Canada
Trang N., age 13 - Caroline Davis Intermediate School, San Jose, CA
Ramjotvir S., age 13 - Caroline Davis Intermediate School, San Jose, CA
Angela V., age 12 - Caroline Davis Intermediate School, San Jose, CA

View most of the solutions submitted by the students above