The Importance of Visualizationin Problem Solving

Pat and Quincy run a 100-meter race. Pat wins by 10 meters. To even up the race, they decide to run again, but this time Pat begins 10 meters behind the start line. Assuming they run at the same constant speed as before, who wins?

Problem 1 - The Foot Race

Pat still wins.

Quincy wins.

It is a tie.

Pat

Quincy

Pat

Quincy

Fin

ish L

ine

still 10 meters to go...

What is the probability that the next person you meet has an above average number of arms?

Certain.

Very likely.

Likely

Unlikely

Very Unlikely

Impossible

There are more people with less than 2 arms thanthere are with more than two arms.

Therefore the average number of arms is < 2...

A cubic block of wood is cut into 27 smaller cubes as shown below using 6 straight cuts. Can this be accomplished in fewer straight cuts by cutting and stacking the pieces before the next cut?

Paint the outside of the large cube...

Some of the small cubes will have paint on 1, 2, or 3 surfaces...

One of the cubes will have no paint...

Can 31 dominoes be arranged to exactly cover the 62 squares of a checkerboard with two of it's opposite diagonal squares removed as shown?

What does one domino cover regardless of its position?

How many squares of each color are there on the modified checkerboard?

Monty Hall'sLet's Make a Deal

Monty Hall, the host of a famous TV game show, has a player pick one of three doors (or curtains). A great prize is behind one of the doors and joke prizes behind the other two. Once the player selects a door, Monty reveals a joke prize behind one of the doors not picked, leaving two doors. Then Monty asks the player if they want to switch to the other door. Should the player switch?

It's better for the player to keep the door they selected.

It's better for the player to switch to the other door.

It doesn't affect the player's chances of winning either way.

For this problem, it may help to visualize a second player that gets the door the first player gives up. Since Monty is revealing a joke prize one of the two players will always win.

When player one first picks a door the chances of picking the great prize is 1/3.

If player one doesn't switch their odds don't change, and since one of the two players is guaranteed the great prize, the odds for player two is 2/3.

Three people sit down to eat together, one has 5 loaves, the second has 3 loaves and the third has 8 coins but no loaves. The one with the coins offers to give all of them for an equal share of the loaves. How many coins should each of the others get?

How much did each contribute?

Erin travels from home to a weekend getaway in heavy traffic. Leaving the house at exactly 7:00 a.m. and arriving at half past noon. Driving was erratic with occasional stops.

For the trip home, Erin leaves again at exactly 7:00 a.m. but this time traffic is light and driving is steady, so the trip takes 1 hour less time.

How likely is it that Erin was at the same point at exactly the same time on both days of travel?

Certain.

Very likely.

Likely

Unlikely

Very Unlikely

Impossible

7:00 a.m. 12:30 p.m.

7:00 a.m.11:30 a.m.

7

8

9

10

11

12

1

Distance

Tim

e

B

E

F

A C D

Can you move just one coin to make two lines containing four coins each?

How about now?

no, you can't useany of these...

How about now?

no, you can't useany of these...

You travel from here to there, a trip of 150 miles averaging 40 mph. Then you travel back along the same route and the same distance of 150 miles at an average of 60 mph. What is the average miles per hour for the entire 300 mile trip?

What is the time of the trip from here to there?... _________________

What is the time of the trip from there to here?... _________________

What is the total time of the round trip ............... _________________

What is the total distance traveled? ................... __________________

What is the average speed?............................... ___________________

In this case, you should gather the essential data step-by-step...

150 mi/40 mph = 3.75 h

150 mi/60 mph = 2.50 h

6.25 h

300 mi

300mi/6.25h = 48 mph

1.

2.

3.

4.

5.

Move three balls to reverse the direction of the triangle...

Sometimes it helps to look for symmetry..

You are taking a trip on an interstate highway. You know the mile-marker for your exit, say it's mile 1. You are traveling south in the same state as your exit so you are passing mile-markers with values decreasing toward your destination.

Describe a simple way to estimate your remaining time of the trip (without a GPS navigator) assuming you are traveling at a constant 75 mph.

Calculate the time it takes to travel 1 mile.

1 hr / 75 mi = 60 minutes / 75 miles = 0.8 minutes/mile

Now you can simply multiply the miles remaining by 0.8 to find minutes r

For example 90 miles remaining = 72 minutes to go.

Some handy ratios are:

60 mph = 1 minute per mile

67 mph ~ .9 minutes per mile

75 mph = .8 minutes per mile

80 mph = .75 minutes per mile

You live 15 miles from your workplace. One morning youoversleep and leave the house late. The speed limitbetween your home and work is 55 mph, but you decide torisk a speeding ticket by driving 75 mph. How much timedo you make up?

15 miles/55 mph x 60 min/hr = 16.36 minutes

15 miles/75 mph x 60 min/hr = 12.00 minutes

You save < 5 minutes

20 mph over the speed limit is $150 + court cost and 3 points off license

This was a Public Service Announcement

A street vender was asked how many chickens he had sold at market that day. His reply was:

I've had four customers today, and each boughthalf of my remaining chickens, plus a half chicken.

If the vender sold all of his chickens at market that day, how many chickens did he sell?

http://www.brainbashers.com/

In this case it helps to visualize the problem in reverse.

1. If half the chickens plus 1/2 a chicken is all the chickens,how many are there?

From this point forward we simply add half a chicken and then double the number of chickens.

2.

3.

4.

Pat was supposed to take two pills, one each of two different prescriptions: Xyloprine, and Yakzolate. Unfortunately, two pills fell out of the Yakzolate bottle and got mixed up with a Xyloprine pill. Even worse, they look exactly the same. How can Pat make sure to take one of each pill without wasting any medicine?

In this case, it helps to recognize that there is no way to remove the wrong pill, and the current dosage has one too many Yakzolate pills...

Since this pile contains one Xyloprine and two Yakzolate...

My mother purchased 100 pounds of potatoes last week, which I knew to be exactly 99% water. After a week, drying in the sun, they had dried a little to become only 98% water. What was the total weight of the potatoes now?

This one is just weird no matter how you visualize it...

If 100 pounds of potatoes is 99% water then they are 1% potato by weight.

If they lose enough water so that the 1 lb of potato represent 2% of thetotal weight (i.e. 98% water)...

Then the total weight must now be 50 times the weight of the potato or 50 lbs.

100

90

80

70

60

50

40

30

20

10

% o

f To

tal W

eigh

t

10 20 30 40 50 60 70 80 90 100

Total Weight of Water (lbs)