X.  Deductive and Hypothetical Thinking through Days of the Week

A businessman had to work late and did not arrive home until early morning for several days, but his wife suspected him of being out with a girlfriend. One early morning he came home and found this note:

The humorous effect of this note comes from the shifts of perspective in time. The last sentence, for example, uses the fact that by tomorrow, today will be yesterday.

The mental activity involved in following descriptions of this sort—comprehending verbal statements of movements along some dimension and reversing movements by thinking backward through a sequence to see where a movement began—is fundamental in solving many types of mathematics problems. The exercises in this chapter strengthen your ability to follow and graphically represent complicated descriptions, and to change perspectives in reversing operations. Try this sample exercise.

The following diagram shows that 1 day after Tuesday is Wednesday; and 2 days after Tuesday is Thursday.

Therefore, my birthday is on Thursday.

Part I

Each of the following problems is accompanied by a diagram. Show on the diagram all the steps you use to arrive at the answer. Although you may be able to solve the easier problems without a diagram, you will find the diagram indispensable for the more difficult problems that come later.

Exercises

1.  Suppose Valentine’s Day is 3 days after Friday. What day is Valentine’s Day?

2.  Suppose Lincoln’s birthday is 4 days before Thursday. What day is Lincoln’s birthday?

3.  Suppose Christmas is 2 days after Wednesday?

a.  What day is Christmas?

b.  Based on your answer to Part a, what day is 4 days before Christmas?

Part II

The next problems are trickier because you must work backward from the given information. Try this sample problem before checking the answer below.

Friday is 2 days after Easter. What day is Easter?

Here are the steps you could use to solve the problem. First the given information is represented on the diagram.

Because Friday is after Easter, we know Easter must be earlier in the week. Thursday is 1 day after Easter.

Therefore, Wednesday must be Easter.

Use similar steps in solving the following problems.

Exercises

1.  Wednesday is 3 days after Halloween. What day is Halloween?

2.  Saturday is 5 days before Labor Day. What day is Labor Day?

3.  The two parts of this exercise contrast the two types of problems you have been solving.

a.  Suppose Christmas is 2 days after Thursday. What day is Christmas?

a.  Suppose Thursday is 2 days after Christmas. What day is Christmas?

Part III

Often in mathematical or other logical problems, you must assume something is true and then deduce conclusions that follow. Here are four ways of presenting information you are asked to assume in deducing a simple conclusion.

Assume today is Tuesday. What is tomorrow?

Suppose today is Tuesday. What is tomorrow?

If today is Tuesday, what is tomorrow?

Today is Tuesday. What is tomorrow?

In reality, today may be Thursday and tomorrow Friday. But each of these four questions asks you to assume today is Tuesday, in which case tomorrow is Wednesday. For the following exercises, assume the given information is true and then deduce the answer requested. Try this problem before checking the answer.

Today is Wednesday. What is 4 days after tomorrow?

This problem can be easily solved by labeling Wednesday as today on a diagram, and then counting off the appropriate days to find the answer.

Exercises

For each exercise, make your own diagram and show the steps you use to reach your answer.

1.  Today is Thursday. What is 2 days after tomorrow?

2.  Today is Friday. What is 6 days before yesterday?

3.  Yesterday was Monday. What is 4 days after tomorrow?

4.  Today is Saturday. What is the day after 4 days before tomorrow?

Part IV

Generally, complex mathematical problems are solved by breaking them into parts and working step by step. Consider this problem.

One way to begin solving such a problem is to separate it into parts like this.

Now we can use a diagram to work step by step through the problem.

Step 1.  We are told today is Monday.

Step 2.  This means yesterday was Sunday.

Step 3.  Now we need to find 3 days before yesterday. One day before yesterday was Saturday; 2 days before yesterday was Friday; so 3 days before yesterday was Thursday.

Step 4.  Finally, we need to find 1 day after this.

Note how the four steps correspond to four parts of the original problem.

Use a similar step-by-step approach in solving the following problems.

Exercises

For each exercise, make your own diagram and show the steps you use to reach your answer. You might find it useful to mark each problem into parts, as shown in the first problem.

1.   Today is Friday. What is 2 days before 5 days after yesterday?

2.   Today is Thursday. What is 6 days before 3 days after tomorrow?

3.   Yesterday was Tuesday. What is 2 days before 4 days after tomorrow?

4.   Tomorrow is Sunday. What is 2 days after 3 days before yesterday?

5.   Today is Tuesday. What is 2 days after 10 days before the day after tomorrow?

6.   Yesterday was Saturday. What is 4 days before 7 days after 2 days before today?

7.   You can easily make up new problems by circling alternatives in this problem generator.

Here is an example:

a)  

b)  Generate a problem by circling different alternatives. Then solve the problem, showing all your steps on a diagram.

8.   More complicated problems can be created by adding more moves. See if you can analyze this one by separating the moves and working through them one after another. Use a diagram.

Today is Monday. What is 3 days after 2 days before 6 days after 5 days after tomorrow?

Part V

The next set of problems are different from the last set, just as the problems you solved in Part I were different from those in Part II. To see the difference, solve these two problems before checking the answers.

A.  Today is Sunday. What is 3 days after today?

B.  Sunday is 3 days after today. What is today?

Both problems contain the word after. But in Problem A you move to the right to find the answer.

For Problem B you move to the left to find the answer because Sunday is already after the answer.

Problems like B are generally more difficult because you must reverse your thinking process. When you see the word after, you generally do not look for the answer later in the week but earlier—the day mentioned in the assumed information is already after the answer. Similarly, when you see before, you generally do not look for the answer earlier in the week but later. Try this problem, labeling the diagram fully, before checking the answer.

Sunday is 3 days before yesterday. What was yesterday?

Here are the steps you can use to solve the problem.

Step 1.  Label the day on the diagram

Step 2.  Is Sunday before or after yesterday? __before__

Step 3.  Which direction is yesterday? __→__

Step 4.  Use the diagram to complete the problem.

If you have difficulty with any of the following problems, review the simpler but similar problems in Part II. When mathematicians encounter difficulty with a problem, they often examine similar but simpler problems to see how the problems are solved.

Exercises

1.  Sunday is 4 days after yesterday. What is yesterday?

a.  Label the day on the diagram.

b.  Is Sunday before or after yesterday? ________

c.  Which direction should you move to find yesterday? ________

d.  Find and label yesterday on the diagram.

2.   Tuesday is 2 days before tomorrow. What is tomorrow?

a.  Label the day on the diagram.

b.  Is Tuesday before or after tomorrow? ________

c.  Which direction should you move to find tomorrow? _________

d.  Find and label tomorrow on the diagram.

3.   Thursday is 3 days after tomorrow. What is today.

a.  Label the day on the diagram.

b.  Is Thursday before or after tomorrow? __________

c.  Which direction should you move to find tomorrow? __________

d.  Find and label tomorrow on the diagram.

e.  Find and label today on the diagram.

4.   Friday is 3 days before yesterday. What is tomorrow?

a.  Label the day on the diagram.

b.  Is Friday before or after yesterday? ________

c.  Which direction should you move to find yesterday? _________

d.  Find and label yesterday on the diagram.

e.  Label today on the diagram.

f.  Label tomorrow on the diagram.

Part VI

The next problems involve two moves. Try this problem before checking the answer.

Monday is 5 days before 2 days after yesterday. What is yesterday?

Here are the steps you can use to solve this problem.

Step 1.  Separate the problem into two moves

What is yesterday?

Step 2.  Label the day on the diagram.

Step 3.  Is Monday 5 days before or after part 2? __before__

Step 4.  Which way must you move first? __→__

Step 5.  Use the diagram to make the first move. Label the diagram.

Step 6.  Which direction must you move in Part 2 to find yesterday? __←__

Step 7.  Use the diagram to complete the problem. Label your moves.

Step 8.  Check your answer by starting with Thursday as yesterday and reversing the steps to see if you reach Monday.

Exercises

1.   Monday is 4 days before 1 day after tomorrow. What is tomorrow?

a.  Separate the problem into two moves.

b.  Label the day on the diagram.

c.  Is Monday 4 days before or after Part 2? _________

d.  Which way must you move first? _________

e.  Use the diagram to make the first move. Label the diagram.

f.  Which direction must you move in Part 2 to find tomorrow? ___________

g.  Use the diagram to complete the problem.

h.  Check your answer.

2.   Thursday is 3 days after the day before yesterday. What is yesterday?

a.  Separate the problem into two moves.

b.  Label the day on the diagram.

c.  Is Thursday before or after Part 2? __________

d.  Which way must you move first? _________

e.  Use the diagram to make the first move.

f.  Which direction must you move in Part 2 to find yesterday? _________

g.  Use the diagram to complete the problem.

h.  Check your answer

Include a fully labeled diagram that shows all your steps in solving the remaining problems.

3.   Wednesday is 6 days before 2 days after tomorrow. What is tomorrow?

4.   Monday is 3 days before 2 days before today. What is tomorrow?

5.   Sunday is 2 days after 6 days before tomorrow. What is today?

6.   Saturday is the day after 3 days after yesterday. What is today?

7.   Make up a problem of this type, then solve it showing all your steps on a diagram. Turn it in to your instructor. If you do not have time to complete it in class, do it for homework.

Part VII

The problems you have been solving are complicated counting problems: You count through the days of the week to get the answer. Mathematics has its roots in counting—counting events, objects, and distances—so these are very basic mathematics problems. They require you to think analytically, deductively, graphically, and sometimes in “reverse,” the same forms of thinking used in all advanced mathematics.

The exercises in this last section involve both types of problems you have been solving, along with other operations. Use a diagram to solve the “days of the week” portion of each exercise.

Exercises

1.   Yesterday was Friday. What is the third letter in the day after tomorrow?

2.   If yesterday was Tuesday, is the third letter of 2 days after tomorrow in the first or second half of the alphabet?

3.   If 2 days after tomorrow will be Sunday, what position in the alphabet is the first letter of the day before yesterday?

4.   If 3 days after yesterday is Saturday, how many letters before z in the alphabet is the first letter of 4 days before tomorrow?

5.   If yesterday was Monday, are there more letters in 2 days after tomorrow or 3 days after tomorrow?

6.   If 6 days ago was Wednesday, what is the second letter after the second letter in 2 days after tomorrow?

7.   If 6 days after tomorrow is Tuesday, what is the day before 3 days after tomorrow?

8.   If 5 days before the day before tomorrow is Saturday, what is 2 days after today?

9.   If the day after 3 days after tomorrow is Wednesday, what is 3 days before 6 days after yesterday?

10.  If the day before 3 days after tomorrow is Wednesday, what is 2 days after 5 days before yesterday?

11.  

Select a set of alternatives that generates the easiest possible problem for you to solve. Then solve it, showing your steps on a diagram.

12.  Select a set of alternatives which generates the hardest possible problem for you to solve. Then let your partner solve it, showing his work on a diagram.

13.  Devise a problem generator like the one in Problem 11, but one that generates harder problems. Use it to generate the hardest problem it can. Then let your partner solve the problem, showing his work on a diagram.

14.  Make up a problem like 4. Then let your partner solve it, showing his steps on a diagram.

Extra Practice Problems

1.   Wednesday is 5 days before 3 days after tomorrow. What is tomorrow?

2.   Tuesday is 4 days before 2 days before today. What is tomorrow?

3.   Friday is 2 days after 4 days before tomorrow. What is today?

4.   Sunday is the day after 4 days after yesterday. What is today?

5.   Friday is the day before 3 days before yesterday. What is tomorrow?

The skills of working step by step and making diagrams that you have learned from this chapter have many applications. In chapter 11, you see how they can be applied to solving mathematical word problems. Step by step reasoning is more important today, in the computer age, than it has ever been. But it has always been the keystone for logic of all kinds. The following three problems are from G. A. Wentworth’s 1895 text Arithmetic. Approach them in the same manner as you did the earlier problems: work step by step toward your answer and use scratch work or diagrams when helpful.

6.   If a workman has taken every day, for the last 12 years, two glasses of beer at 5¢ a glass, how much could he have saved if he had not indulged this habit, reckoning 365 days each year?

7.   A man divides $1622.50 among four persons so that the first has $40 more than the second, the second $60 more than the third, and the third $87.50 more than the fourth. How much did the fourth person receive?

8.   A man bequeathes to his wife of his estate: to his daughter, of it; to his son, of the daughter’s share: he divides the remainder equally between a hospital and a public library. What part is received by the hospital?