Author:
Amanda Hager
Subject:
Mathematics
Material Type:
Homework/Assignment, Lesson
Level:
Academic Lower Division
Provider:
University of Texas at Austin
Tags:
  • Mindset
  • Not Reviewed
  • d2s2
  • mindset
  • License:
    Creative Commons Attribution
    Language:
    English
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    Education Standards

    1.2 Puzzles

    1.2 Puzzles

    Overview

    TCCNS CourseMATH 1332: Contemporary Mathematics
    UT Austin CourseM 302: Introduction to Mathematics

     

    Introduction

    Suggested Resources and Preparation

    Materials and Technology

    • For the instructor: either a computer/projector or a blackboard/whiteboard.

    • Optional for the instructor: manipulatives in the form of homogenous objects (like marbles or beans) and containers to represent the scales (pairs of cups or rings made of pipe cleaners).

    • For the student: blank paper or a word processor is sufficient.

    Prerequisite Assumptions

    This exploration-based lesson required no previous experience or knowledge. In a university environment where students often add the course late, can't find their lecture hall, or have been away from math courses for multiple years, it is helpful to have these "puzzle days" where students are working through content that is relevant yet not critical.

    Lesson Length

    50 minutes

    Overview

    This exploration-based lesson required no previous experience or knowledge. Students are briefly introduced to the concepts of cognitive bias and listing cases when solving problems. Two puzzles are discussed in detail, then students solve related and similar puzzles.

    Lesson Objectives

    You will understand that:

    • Some problems can be more easily solved if we systematically list all possible cases.

    You will be able to:

    • List all cases when solving quantitative problems or puzzles.
    • Arrange cases in a logical way and clearly explain the consequences that are implied by each case being true.

     

    Orange Juice Puzzle

    glass of orange juice, decorative image

    The Puzzle

    Imagine that you are selling orange juice. You have a very large container of juice, say a whole barrel of it, and you have two sizes of cups available. Small cups are 6 oz and large cups are 10 oz.

    A very picky customer comes up and demands precisely 8 oz. of juice in a 10 oz cup. You are not allowed to mark the cups in any way, and it is impossible to "eyeball" the amount of juice (the glasses are not cylinders). Also, for some random reason, you only get to use one of each size of cup to pull it off. Is this possible?

    Question: Do you think it will be possible to give this customer exactly 8 oz of juice in the 10 oz cup? Give a reason why this was your first instinct.

    Exploring Possibilities, Getting Stuck

    Did you by chance get a little stuck on the idea of marking the cups or estimating the amount of liquid in the cups? If you did, is there any possibility that you got stuck on those things because I said "no marking the cups" and "no eyeballing volume?" If so, you may have fallen victim to anchoring bias. The anchoring bias is the tendency to fixate on early pieces of information we get about problems, and those early pieces of information influence our decision making.

    Let's suppose your instructor designed two different two-question quizzes and gave each quiz to exactly half of the students in your class:

    Quiz Version 1

    1. Is the height of the tallest redwood more or less than 1,200 feet?
    2. What is the height of the tallest redwood? Make your best guess.

    Quiz Version 2

    1. Is the height of the tallest redwood more or less than 180 feet?
    2. What is the height of the tallest redwood? Make your best guess.

    Do you feel your answers to #2 in both quizzes would be equal? A 1982 study1 of visitors to the San Francisco Exploratorium found that the average (mean) guess for the first group was 844 feet, and the average for the second group was 282 feet. Reading the first question influenced the responses to the second question. Even if the things we read and hear are unrelated, they can influence our thoughts and decisions.

    Beating Anchor Bias

    As with any type of bias, the first step is to acknowledge it happens. Taking a more mindful and reflective approach to decision making, problem solving, and negotiation can help you think more clearly. Were you in a hurry when you read the puzzle? Were you anxious?

    Second, if you got stuck, stop what you are doing, take a step back, and examine the beliefs and assumptions you have about the problem and where they came from. If you thought to yourself "It's impossible to do this without marking the cups," then perhaps this came from the belief "I need to mark the cups" which came from the mention early on that you are not allowed to mark the cups.

    Hint

    So back to the problem: nothing is preventing you from pouring the contents of a full cup back into the barrel. Maybe it's a little weird or unhygienic, but you can do it. And you can also transfer juice from one cup to the other. So what happens if you fill the big cup all the way and pour juice into the small cup until it's full? You get four oz in the large cup and 6 oz in the small cup:

    decorative image, large glass with 4 oz of liquid, small glass completely full with 6 oz of liquid

    Now let's dump the 6 oz out of the small cup into the barrel and transfer the 4 oz from the big cup to the small cup:

    decorative image, empty large glass, 4 oz in the small glass

    That's half of what the picky person wanted.

    Question: What do you think we should try next?

    Dumping the small cup just gets you back to where you started, and if you fill the small cup the rest of the way to the top, why bother transferring juice to the small cup at all? The thing to do is fill the big cup a second time and carefully transfer more juice to the small cup until it's full. Then the big cup will have exactly 8 oz of juice in it!

    decorative image, rest of solution shown, large glass is filled, two ounces is transferred to small glass, 8 oz remains

    Magic Stones Puzzle

    The puzzle

    Magic Stones puzzle description

    You have nine identical-looking stones. The valuable stone that you want is a tiny bit heavier than the others. All you have to figure out which one is heavier are two old-fashioned scales that can be used one time each. You can load up both sides of a scale with stones, and the scale will tell you which side is heavier. That's it.

    two scales. left scale is uneven, right scale is even

    Question: Do you think it is possible to figure out which of the nine stones is heavier using just two scales?

    Examining assumptions

    In a live class, the most common response to this question is "If I load up four stones on each side of a scale, then either the two sides equal and I win, or one side is heavier and I'm down to four stones. I can then put two on each side of the second scale, and I know one side will be heavier. But that's only down to two stones." And that's pretty good, but we really want that money. In this case, you have to go back and examine the choices you made so that you can try something else.

    In this case with the fictional student above, there were two choices made: they chose four and four, and then they chose two and two. One of those choices needs to be different. It turns out that there aren't that many possibilities, since you can't have uneven amounts: you can take one, two, three, or four on each side. Four didn't work out. And one seems pretty ridiculous. What about two or three?

    Using cases

    So let's say you load up three stones on each side of the first scale, so that three stones are left off the scale. 

    Question: What are all of the possible outcomes of this weighing?

    Either one side of the scale is heavier, or both sides are the same. These are the only two possibilities, or cases. If we can logically list every single possible case, and there aren't too many, we can write really clear arguments and save time doing it.

    In this case, if one side of the scale is heavier, then we have narrowed it down to three stones.

    In the other case, if both sides of the scale are equal, then the heavy stone is off the scale and we have narrowed it down to three stones!

    case 1: heavy stone must be on the scale. uneven scales shown with three stones per side, three left off to the side. Case 2: heavy stone must be off the scale. Even scales shown with three stones per side, three stones left off to the side.

    No matter what, we're down to three stones. Can you see what to do next?

    The only real possibility is to load up one stone on each side of the second scale. Case 1: if one side of the scale is heavier, you know which one is valuable. Case 2: if both sides of the scale are equal, then the valuable stone is off the scale.

    Quiz Questions

    Solutions

    QuestionChoice
    13
    24
    32

    Question 1

    The anchoring bias is:

    1. our tendency to get stuck on a conclusion and not listen to any arguments.
    2. our tendency to be unable to proceed on problems we are solving because they are too difficult.
    3. our tendency to focus too much on early pieces of information when we are solving problems or making decisions.
    4. our tendency to seek out evidence that supports the claim we want to make and ignore evidence that refutes it.

    Question 2

    How can you use one single-use scale to tell which one of 3 identical-looking stones is slightly heavier than the others?

    1. It is impossible to do since there are an odd number of stones and you can only use the scale one time.
    2. Put two stones on one side and one stone on the other side. The scale will tell you which side the heavy stone is one.
    3. The best you can do is put one stone on each side of the scale. If the scale is uneven, you know the heavy stone is on the scale. If the scales are even, you are out of luck.
    4. Place one stone on each side. If the scale is uneven, then the heavy stone is on the scale. If the scale is even, then the heavy stone is off the scale.

    Question 3

    If you have a 9 ounce cup and a 6 ounce cup and you wanted to measure out exactly 3 ounces of juice into the 9 ounce cup without estimating, how can you acheive this?

    1. Eyeball the 1/3 level in the 9 ounce cup and fill the cup 1/3 of the way.
    2. Fill up the 9 ounce cup all the way and then transfer juice into the 6 ounce cup until it is completely full. There will be 3 ounces left over in the larger cup.
    3. Fill up the 6 ounce cup halfway by estimating half of the height of the cup and then pouring that juice over to the larger cup. 
    4. Fill up the 6 ounce cup all the way and then transfer juice over to the larger cup until the volumes look similar.

    Homework Questions

    Instructor Solutions

    Question 1.2.1

    Yes it is possible. Anytime there is one solution there will always be a second. In table form:

    MoveLarge CupSmall cup
    Fill small cup06
    Transfer small to big60
    Fill small cup66
    Transfer small to big until full102
    Dump big cup02
    Transfer small to big20
    Fill small cup26
    Transfer small to big80

    Question 1.2.2

    You have a very large container of juice, say a whole barrel of it, and you have two sizes of cups available. Small cups are 5 oz and large cups are 8 oz. A very picky customer comes up and demands precisely 7 oz. of juice in a 8 oz cup. You are not allowed to mark the cups in any way, and it is impossible to "eyeball" the amount of juice. Is this possible?

    Yes. Two solutions shown.

    MoveLarge 8 oz cupSmall 5 oz cup
    Fill large cup80
    Transfer large to small until full35
    Dump small cup30
    Transfer large to small03
    Fill large cup83
    Transfer large to small until full65
    Dump small cup60
    Transfer large to small until full15
    Dump small cup10
    Transfer large to small 01
    Fill large cup81
    Transfer large to small until full45
    Dump small cup40
    Transfer large to small04
    Fill large cup84
    Transfer large to small until full75
    MoveLarge 8 oz cupSmall 5 oz cup
    Fill small cup05
    Transfer small to large50
    Fill small cup55
    Transfer small to large until full82
    Dump large cup02
    Transfer small to large20
    Fill small cup25
    Transfer small to large70

    Question 1.2.3

    Several solutions exist:

    1. Load four and four. One side will be heavier; take those four and load two and two on second scale. One side will be heavier; take those four and load one and one on last scale.
    2. Load three and three, with two to the side. If one side is heavier, take those three and load the second scale with one and one with one to the side. If the sides of the first scale are even, take the two off to the side and split them on the second scale.
    3. Load four and four, then one and one, then one and one on the third scale if necessary.
    4. Load two and two. If the sides are uneven, split the two heavy stones on the second scale. If the sides are even, then split the four others on the second scale and split the heavier two on the third scale.
    5. Load one and one. If the sides are uneven, you win. If the sides are even, load two and two on the second scale with two off to the side (using the third scale if necessary).

    Question 1.2.4

    Load 9 stones on each side of the first scales with 9 off to the side. No matter what, you'll reduce from 27 to 9 possible stones. Load the second scales with 3 and 3, with 3 off to the side. You'll be down to three possible stones. Load the third scales with 1 and 1, with 1 off to the side.

    Question 1.2.5

    Solution 1: Use three cases based on where the money is:

    1. If the money is in box 1, then signs 1 and 2 are false and sign 3 is true. This is possible.
    2. If the money is in box 2, then signs 1 and 2 are true and sign 3 is false. This is impossible.
    3. If the money is in box 3, then signs 1 and 3 are true and sign 2 is false. This is impossible.

    Solution 2: Use three cases based on which sign is true:

    1. If sign 1 is true and signs 2 and 3 are false, then the money is not in box 1, is not in box 2, and is in box 2. This is impossible.
    2. If sign 2 is true and signs 1 and 3 are false, then the money is in box 1, is in box 2, and is in box 2. This is impossible.
    3. If sign 3 is true and signs 1 and 2 are false, then the money is in box 1, is not in box 2, and is not in box 2. This is posslble.

    In both situations, we see only one possibility. The money is in box 1, and sign 3 is true.

    Question 1.2.6

    As in the lesson, let's say you have a 6 ounce cup and a 10 ounce cup. Fill the big cup with juice and transfer it to the small cup repeatedly, dumping the small cup out as necessary.

    1. What amounts are you able to produce in the larger cup?
    2. What happens if you keep repeating the procedure?
    MoveLarge CupSmall cup
    Fill large cup100
    Transfer large to small until full46
    Dump small40
    Transfer large to small04
    Fill large cup104
    Transfer large to small until full86
    Dump small80
    Transfer large to small until full26
    Dump small 20
    Transfer large to small02
    Fill large cup102
    Transfer large to small until full66
    Dump small60
    Transfer large to small06
    Dump small00
    Fill large cup100

    At this point, we are back where we started. The table just goes in loops. That means we can generate these amounts in the large cup: 2, 4, 6, 8, and 10 ounces. (Students may also include 0, depending on how they feel about 0 ounces being an amount of something).

    Question 1.2.7

    As in Question 1.2.2, let's say you have a 5 ounce cup and an 8 ounce cup. Fill the big cup with juice and transfer it to the small cup repeatedly, dumping the small cup out as necessary.

    1. What amounts are you able to produce in the larger cup?
    2. What happens if you keep repeating the procedure?

    You are to produce any integer amount of juice in the larger cup, from 1 (or zero) ounce to 8 ounces. If you keep the table going, the amounts will eventually loop. (This is happening becuase 5 and 8 are relatively prime).

    Question 1.2.8

    Mindset question. Responses vary.

    Question 1.2.1

    In the lesson, where the cups were 10 oz and 6 oz and we wanted 8 oz, we began by filling the larger cup with juice. What happens if you begin by filling the smaller cup with juice and transferred it to the larger cup repeatedly? Can you eventually get 8 oz of juice in the larger cup?

    Question 1.2.2

    You have a very large container of juice, say a whole barrel of it, and you have two sizes of cups available. Small cups are 5 oz and large cups are 8 oz. A very picky customer comes up and demands precisely 7 oz. of juice in a 8 oz cup. You are not allowed to mark the cups in any way, and it is impossible to "eyeball" the amount of juice. Is this possible?

    Question 1.2.3

    Explain how you could use three single-use scales to tell which one of eight identical-looking stones is slightly heavier than the others.

    Question 1.2.4

    Explain how you could use three single-use scales to tell which one of 27 identical-looking stones is slightly heavier than the others.

    Question 1.2.5

    There are three boxes. Exactly one box has $10,000 in it; the others are empty. Each box has a sign behind it with a statement. You are told that exactly one sign is true and the other two are false. The sign behind the first box reads "The money is not in this box." The sign behind the second box reads "The money is in this box." The sign behind the third box reads "The money is not in the second box." 

    Which box has the money?

    Decorative image

    Question 1.2.6

    As in the lesson, let's say you have a 6 ounce cup and a 10 ounce cup. Fill the big cup with juice and transfer it to the small cup repeatedly, dumping the small cup out as necessary.

    1. What amounts are you able to produce in the larger cup?
    2. What happens if you keep repeating the procedure?

    Question 1.2.7

    As in Question 1.2.2, let's say you have a 5 ounce cup and an 8 ounce cup. Fill the big cup with juice and transfer it to the small cup repeatedly, dumping the small cup out as necessary.

    1. What amounts are you able to produce in the larger cup?
    2. What happens if you keep repeating the procedure?

    Question 1.2.8

    Think of a time in your life where it was helpful or would have been helpful to systematically list cases. Often, this will be a time when you needed to prove a point or make a decision. Describe the situation and then list the cases that you used or would have used.