working backwards problem solving example

Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Working Backwards

As I’ve shared before, there are many different ways to go about solving a math problem, and equipping kids to be successful problem solvers is just as important as teaching computation and algorithms . In my experience, students’ frustration often comes from not knowing where to start. Providing them with strategies enables them to at least get the ideas flowing and hopefully get some things down on paper. As in all areas of life, the hardest part is getting started! Today I want to explain how to teach  problem solving by working backwards .

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–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Solve a Math Problem by Working Backwards: 

Before students can learn to recognize when this is a helpful strategy, they must understand what it means. Working backwards is to start with the final solution and work back one step at a time to get to the beginning.

It may also be helpful for students to understand that this is useful in many aspects of life, not just solving math problems.

To help show your students what this looks like, you might start by thinking about directions. Write out some basic directions from home to school:

• Start: Home
• Turn right on Gray St.
• Turn left on Sycamore Ln.
• Turn left on Rose Dr.
• Turn right on Schoolhouse Rd.
• End: School

Ask students to then use this information to give directions from the school back home . Depending on the age of your students, you may even want to draw a map so they can see clearly that they have to do the opposite as they make their way back home from school. In other words, they need to “undo” each turn to get back, i.e. turn left on Schoolhouse Rd. and then right on Rose Dr. etc.

In math, these are called inverse operations . When using the “work backwards” strategy, each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backwards they will need to subtract. And if they multiply working forwards, they must divide when working backwards.

Once students understand inverse operations , and know that they must start with the solution and work back to the beginning, they will need to learn to recognize the types of problems that require working backwards.

In general, problems that list a series of events or a sequence of steps can be solved by working backwards.

Here’s an example:

Sam’s mom left a plate of cookies on the counter. Sam ate 2 of them, his dad ate 3 of them and they gave 12 to the neighbor. At the end of the day, only 4 cookies were left on the plate. How many cookies did she make altogether?

In this case, we know that the final cookie amount is 4. So if we work backwards to “put back” all the cookies that were taken or eaten, we can figure out what number they started with.

Because cookies are being taken away, that denotes subtraction. Thus, to get back to the original number we have to do the opposite: add . If you take the 4 that are left and add the 12 given to the neighbors, and add the 3 that Dad ate, and then add the 2 that Sam ate, we find that Sam’s mom made 21 cookies .

You may want to give students a few similar problems to let them see when working backwards is useful, and what problems look like that require working backwards to solve.

Have you taught or discussed problem solving by working backwards  with your students? What are some other examples of when this might be useful or necessary?

Don’t miss the other useful articles in this Problem Solving Series:

• Problem Solve by Drawing a Picture
• Problem Solve by Solving an Easier Problem
• Problem Solve with Guess & Check
• Problem Solve by Finding a Pattern
• Problem Solve by Making a List

So glad to have come across this post! Today, word problems were the cause of a homework meltdown. At least tomorrow I’ll have a different strategy to try! #ThoughtfulSpot

I’m so glad to hear that! I hope you found some useful ideas!! Homework meltdowns are never fun!! Best of luck!

This is really a great help! We have just started using this method for some of my sons math problems and it helps loads. Thanks so much for sharing on the Let Kids Be Kids Linkup!

That’s great Erin! I hope this is a helpful method and makes things easier for your son! 🙂

I’ve not used this method before but sounds like a good resource to teach. Thanks for linking #LetKidsBeKids

I hope this proves to be helpful for you!

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Problem solving. work backwards.

A good mathematics problem will not have an obvious solution. We must consider what information we have been given and what we already know, and how these things may relate to the problem. If we are systematic in our thinking this will often lead us to the solution.

If we don’t seem to be making progress, we can:

• Read the problem again;
• Reread the information given and focus on key information that could be useful;
• Use what we already now to reduce the scope of the problem;
• Record work done carefully, so it is easy to retrace steps and to verify or change the method.

Examples of working backwards to tackle a problem

First we will read the two examples and have a quick think about them and then we will look at how working backwards can help us with each one:

Angle Problem

The following diagram shows an isosceles triangle and a square drawn on a straight line. Find the size of angle A:

Fraction Shaded Problem

The following diagram is of a square with four semicircles drawn inside. What faction of the square is shaded?

Worked Solutions

We can start by adding in the angles that we know about in the square, marking each of them as 90 degrees. Then we can think about the isosceles triangle and the properties of it that we know about. Using this we can calculate that each of the base angles in this triangle must be 70 degrees. Once we have marked each of these angles on, it should be clearer to see what angle A is.

We don’t have a formula for finding this specific area and we don’t know any measurements, so we will have to use letters to represent some of the unknown lengths.

Let’s start by looking at one semicircle, the radius of which we will call r:

Using the formula for the area of a circle, we know that the area of this semicircle is ½πr 2 .

But we don’t want the whole semicircle. We just want the petals. The “white” shape that we need to exclude has a strange shape, but we can tackle the problem instead by excluding a triangle within the semi-circle, which will leave us with “half-petals” from which we can easily find the area of the petals:

Because we have already named the radius of the semicircle as “r”, we can find the area of this “white” triangle, because it has base length 2r and height r, so its area is ½ x 2r x r = r 2 .

So the two red half petals on the diagram above have a combined area or ½πr 2 – r 2 . There are four times these in the whole diagram, so the total area shaded is 4(½πr 2 – r 2 ).

The area of the rectangle is 2r x 2r = 4r 2 .

So the fraction we want is 4(½πr 2 – r 2 ) / 4r 2 . This simplifies to five ½π – 1, which is about 0.57.

The following questions can be solved in different ways. The worked solutions provided afterwards are based on the ways suggested above.

29 Questions of increasing difficulty

Worked Solutions to Questions

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Primary 3 Working Backwards & Its Method

Math heuristics for problem solving, primary 3 working backwards & its method, what is working backwards in math.

The scenario occurs when the quantity data is insufficient to work from the beginning . Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward. This strategy is commonly used in math problems that ask you to find a starting value or figure out what happened before a given situation.

How to Solve Math Questions with Working Backwords Method?

Let's take a look at this primary 3 word problem example:.

Watch the tutorial for free!

Sarah had some pens. She bought 34 pens. She then threw away 29 pens as they were spoilt. In the end, she had 64 pens. How many pens did Sarah have at first?

Identify the Concept

We know this is a Working Backwards question as…

Workings Explained

Always remember when we work backwards, everything will be reversed. Example the 2nd sentence – “She bought 34 pens”. We know when we buy things, we will have more. We need to add. However, when we work backwards, instead of adding, we need to subtract.

• We will start drawing the model from the end by drawing a box and label it “End”. Put the end amount “64” in the box.
• Draw arrow to point to the left, draw another box. On top of the arrow write “+29” as “Sarah threw away 29 pens”. Instead of subtract, we need to add. In the box, write “93” (64+29=93).
• Draw another arrow to point to the left, draw another box. On top of the arrow, write “-34” as Sarah bought 34 pens. Instead of adding, we need to subtract. In the box, write “59” (93-34=59). Label the box “At First” or “Before”.

Sarah had 59 pens at first.

We know this is a Working Backwards question as we were not told the number of pens Sarah had but were asked for the number of pens she had at first.

See other problem-solving strategies and methods

Free Math Problem-Solving Lessons!

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What's Problem Sum?

Problem Sum is a term commonly used in the context of math education, particularly in regions that follow the Singapore Math curriculum . It refers to a type of math problem that typically involves multiple steps and requires students to apply various mathematical concepts and strategies to find the solution. These problems are often word problems that describe a real-world scenario, requiring students to read, comprehend, and analyze the situation before applying the appropriate math operations to solve it.

In essence, a Problem Sum combines elements of arithmetic and logical reasoning , challenging students to think critically and use their problem-solving skills. It is a key component in developing a deeper understanding of math concepts and enhancing analytical abilities in students.

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Working backward to solve problems - maurice ashley.

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Imagine where you want to be someday. Now, how did you get there? Retrograde analysis is a style of problem solving where you work backwards from the endgame you want. It can help you win at chess -- or solve a problem in real life. At TEDYouth 2012, chess grandmaster Maurice Ashley delves into his favorite strategy.

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Working backwards

For problems where you need to create a timeline, meet a deadline, or understand the steps needed to complete a task, working backwards can be a great problem-solving process. To do this, imagine that the task has already been completed or the problem has already been solved with a satisfactory outcome, and work backwards to see what steps were taken to reach that result and when those steps needed to be completed by. Check out the example below to see how this technique works.

The working backwards technique in action

Ruth works in the admin department at a hospital and has just been given an important project to work on. They’ve recently started using a new computer system for the storage of medical files at the hospital, and Ruth needs to develop a training course to teach employees how to use the new system.

To help her plan what steps she needs to take to complete the project, Ruth decides to use the working backwards technique. First, she imagines the course in its completed form. She thinks about what information staff members need to learn and how it would need to function for it to be an effective training tool.

Now, Ruth works backwards to think about the steps that need to be taken to get the course to the completed form that she’s imagined. She also assigns a due date to each of these tasks to help keep the project on track. She does this by taking the deadline she has been given by her boss and working backwards. Here’s the plan she comes up with:

Ruth has worked backwards to figure out how much time she can spend on each step. She now has a clear plan of the steps she will take to complete the project, which she can use to measure her progress along the way. Without a plan like this, Ruth might spend too much time on one step, and not be able to complete another important task, like testing the course for bugs, before handing over the finished product.

Here’s another example of how the working backwards technique can be used:

• Do you like to plan things out ahead of time? Why? Why not?
• For example, working out a budget to save for something or planning a social event.

Think about your particular discipline or area of professional interest — what kind of problems or tasks arise in your area that the working backwards approach could help solve?

• For example, in the area of business sustainability, the problem could be product packaging that is damaging to the environment. A Sustainability Manager who is tasked with reducing the impact of a company’s product packaging would first envision the end goal of packaging with minimal environmental impact, then they’d use the working backwards strategy to develop a plan with all the steps in between. These steps could be things like designing a sustainable packaging strategy, sourcing eco-friendly materials, and implementing changes in the manufacturing line. 

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7.3 Problem Solving

Learning objectives.

By the end of this section, you will be able to:

• Describe problem solving strategies
• Define algorithm and heuristic
• Explain some common roadblocks to effective problem solving and decision making

People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

Problem-Solving Strategies

When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them ( Table 7.2 ). For example, a well-known strategy is trial and error . The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

Method	Description	Example
Trial and error	Continue trying different solutions until problem is solved	Restarting phone, turning off WiFi, turning off bluetooth in order to determine why your phone is malfunctioning
Algorithm	Step-by-step problem-solving formula	Instructional video for installing new software on your computer
Heuristic	General problem-solving framework	Working backwards; breaking a task into steps

Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Everyday Connection

Solving puzzles.

Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below ( Figure 7.7 ) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

Here is another popular type of puzzle ( Figure 7.8 ) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Take a look at the “Puzzling Scales” logic puzzle below ( Figure 7.9 ). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

Pitfalls to Problem Solving

Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but they just need to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. Duncker (1945) conducted foundational research on functional fixedness. He created an experiment in which participants were given a candle, a book of matches, and a box of thumbtacks. They were instructed to use those items to attach the candle to the wall so that it did not drip wax onto the table below. Participants had to use functional fixedness to overcome the problem ( Figure 7.10 ). During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

Link to Learning

Check out this Apollo 13 scene about NASA engineers overcoming functional fixedness to learn more.

Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in Table 7.3 .

Bias	Description
Anchoring	Tendency to focus on one particular piece of information when making decisions or problem-solving
Confirmation	Focuses on information that confirms existing beliefs
Hindsight	Belief that the event just experienced was predictable
Representative	Unintentional stereotyping of someone or something
Availability	Decision is based upon either an available precedent or an example that may be faulty

Watch this teacher-made music video about cognitive biases to learn more.

Were you able to determine how many marbles are needed to balance the scales in Figure 7.9 ? You need nine. Were you able to solve the problems in Figure 7.7 and Figure 7.8 ? Here are the answers ( Figure 7.11 ).

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Working Backwards: Heuristic for Problem Solving

Working Backwards is a non-routine heuristic that all pupils learn in primary schools . Many pupils learn this heuristic as early as when they were in primary two. You can easily find this heuristic being included in one of the topics in assessment books .

Though common, this heuristic is in fact one of the toughest in primary Mathematics syllabus . Problem sums that can be solved using Working Backwards are usually wordy with a sequence of events taking place which make them complex. You need endurance and clarity of mind to follow through a series of events that are unfolding in sequence, not to mention in a backward manner.

Without further ado, let’s delve into one typical upper primary problem sum to find out how we can tackle this category of problem sums using the heuristic of Working Backwards .

Xiaoming and Ali were playing a card game using 96 pokemon cards. In the first game, Ali lost 1 5 of his cards to Xiaoming. In the second game, Xiaoming lost 1 3 of his cards to Ali. After the second game, both boys had the same number of pokemon cards. How many pokemon cards did Xiaoming have at first?

Study and Understand the Problem

In this problem sum, there are only two variables – Xiaoming and Ali.

Cards are transferred between Xiaoming and Ali in a series of games, but the total number of cards between them is still the same – internal transfer.

Think of a Plan

There are contextual clues of a typical “Working Backwards” problem sum:

• Final information is given on how a situation ends and you need to find the answer in the beginning.
• There is a focus on sequence of events.

Act on the Plan

Reverse the solution steps by working backwards in a systematic manner using a table. A table will help to organise your working in a more orderly manner and track the steps in sequence.

Final number of cards each person had = 96 ÷ 2 = 48

	XM	A
End	96 ÷ 2 = 48	96 ÷ 2 = 48

Second game

After Xiaoming lost 1 3 of his cards to Ali, he was left with 2 units as 1 unit was won by Ali (Refer to the numerator and denominator). Thus, 2 units = 48

	XM	A
Second game	2 units = 48 1 unit = 24 3 units = 72 XM had 72 cards before second game	96 – 72 = 24 Ali had 24 before second game

After Ali lost 1 5 of his cards to Xiaoming, he was left with 4 units as 1 unit was won by Xiaoming (Refer to the numerator and denominator). Thus, 4 units = 24

	XM	A
First game	96 – 30 = XM had 66 cards at first	4 units = 24 1 unit = 6 5 units = 30 Ali had 30 cards at first

Reflect on my Answer

Work forward with the answer you have.

XM → 66 A → 30

A → of 30 = 6 30 – 6 = 24 XM → 66 + 6 = 72

XM → of 72 = 24 72 - 24 = 48 (correct!) A → 24 + 24 = 48 (correct!)

More Examples of Problems Sums Involving Working Backwards

Try to pick out the contextual clues that tell you that Working Backwards can be used.

P3 Math question

Alan bought some fish. One day, 6 of his fish died. After that, he bought the same number of fish as those which were still alive. He gave away all his fish equally among 8 friends and each friend had 4 fish. How many fish did Alan have at first?

P6 Math question

A MRT train left Bugis station with some passengers. At Lavender station, no passengers alighted and the number of passengers who boarded the train was  1 4 of the original number of passengers in the train. At Kallang station,  2 5 of the passengers alighted and 51 passengers boarded the train. At Aljunied station, 2 3 of the passengers alighted and 24 passengers boarded the train. At Paya Lebar station, all 122 passengers alighted from the train. How many passengers were there when the train left Bugis Station?

Whenever possible, use a table or draw boxes to help you solve Working Backwards questions in an orderly and systematic manner.

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About the Author

Teacher Zen has over a decade of experience in teaching upper primary Math and Science in local schools. He has a post-graduate diploma in education from NIE and has a wealth of experience in marking PSLE Science and Math papers. When not teaching or working on OwlSmart, he enjoys watching soccer and supports Liverpool football team.

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System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Chapter 3: working backwards.

The very name of this strategy sounds confusing to most people. It is a very unnatural way of doing things. When most of us went to school, we were taught to solve mathematical problems in a direct, straightforward manner. And yet, working backwards is the way many real-life problems are often resolved. As a simple example, suppose you had to pick up your child from football practice at exactly 5:00 p.m. At what time should you leave? Well, let us say it take 30 minutes to get to the ballpark. We would better leave a 5-minute safety valve. Okay, then we need to leave 35 minutes earlier, or not later than 4:25 p.m. Without even thinking about it, we were working backwards! Of course, this is a very simple example of this strategy…

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