
ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary 
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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:
 Buy a package of zoogs. What are the fewest number and types of coins needed to purchase a package of zoogs using exact change?
 Which coin is worth the least, and how many of it does it take to equal each of the other coins?
 How many of the least valuable coins would it take to buy a package of zoogs?
 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?
 Another Earthling arrives on the next shuttle and wants to try some Mushniks. Explain how you figured out the relations between the coins.
 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
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.

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.
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