Eliminate Possibilities

1
Problem Solving Strategy

• Eliminate Possibilities

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Eliminate Possibilities

• Eliminating possibilities is a powerful problem
  solving tool. Detectives and doctors use this
  extensively when solving cases or diagnosing
  illnesses.
• When you use this strategy, consider/list all the
  possibilities and then eliminate those that lead
  to contradictions.
• This is the basis of logical deduction. This
  strategy is often used in combination with other
  strategies to make problem solving more efficient.

3
Eliminate Possibilities

• In addition to eliminating possibilities, it is
  important to remember the possibilities that have
  been eliminated.
• Sometimes not all of the wrong possibilities can
  be eliminated right away.
• Youll need to keep reviewing the clues with each
  case. Here is a diagram that illustrates the
  thinking process used in this strategy.

Assumption doesnt work! It is wrong eliminate
it!
Make assumption Apply the assumption in
the problem
Assumption works! We didnt learn much its
still a possibility!
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Eliminate Possibilities

• You probably apply this strategy when you go to a
  restaurant to eat. You eliminate what you dont
  want to eat on the menu first to narrow down your
  choices.
• When asked to guess a number, you get clues to
  help you eliminate the wrong answers.
• Detectives use this to eliminate possible
  criminals but it can fail to produce the criminal
  if the list of possible suspects is not complete.
  
• Health care professionals must conduct tests to
  diagnose illnesses the tests that come back
  negative are as important as the ones that come
  back positive.

5
Cryptarithmetic Problems

• There are two types of problems in your problems
  for this strategy. This lesson concentrates on
  one type the cryptarithmetic problem.
• Check out this cryptarithmetic problemA message
  hides an arithmetic problem this type of
  problem is perfect for applying this problem
  solving strategy! You see these types of
  problems in many puzzle books!

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Cryptarithmetic Problem
Emil Gordans tomcat and Pearls old nag were
not much use. I traded them for my new goat.

• Olive Hey, that sounds like a good
  cryptarithmetic problem. Lets see if we can
  solve it.
• G T O M
• P N A G
• E G O A T
• Each letter stands for a different digit 5 and
  A is odd.

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Make systematic lists.

0 1 2 3 4 5 6 7 8 9 G
G T O M P N A E 5
8
Generate possibilities.
In the thousands column, 5 P adds to
5. So P 0 and theres no carry to this
column or P 9 and theres a carry of 1 to this
column.
9
Eliminate possibilities by finding contradictions.
The thousands column adds to two digits, so P is
not 0.
P 0 or P 9
10
List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 E G P
G T O M P N A E 5 9 1
Therefore, P 9, theres a carry to this column,
and E 1.
11
Generate possibilities.
In the tens column, O A adds to A. So O 0
and theres no carry to this column or O 9 and
theres a carry of 1 to this column.
12
Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 E G P
G T O M P N A E 5 9 1
P 9, so O ? 9.
O 0 or O 9
13
List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
Therefore O 0 (zero).
14
Generate possibilities.
In the hundreds column, T N 10. So we have
these possibilities for T and N
15
Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
T and N cant be 1, 5, or 9 because these numbers
are already used.
16
Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
T cant be 2, 3, or 4 because M 5 T in the
ones column, and M cant be negative.
17
List the remaining possibilities.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
18
Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
M cant be 1 because E 1.
G T O M P N A E 5 0 9 1
19
Reread the problem.
G T O M
P N A G E G O A
T Each letter stands for a different digit. G
5 and A is odd.
20
Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
M cant be 2 because then T and N would be the
unused odds, leaving none for A.
G T O M P N A E 5 0 9 1
21
List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 O E N M G T P
G T O M P N A E 5 8 0 3 9 2 1
T 8 N 2 M 3
22
Finish the problem and check.
0 1 2 3 4 5 6 7 8 9 O E N M G A T P
G T O M P N A E 5 8 0 3 9 2 7 1
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Strategy Eliminate possibilities

• Generate possibilities.

• Eliminate possibilities by finding
  contradictions.
• List the remaining possibilities.
• Check them against the evidence. There may be
  more than one solution.
• Think about possibilities you might have missed.

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Now its your turn

• Open your problem solving book and go to the
  section on Eliminating Possibilities (pages
  11-12).
• Print off a problem solving sheet and do a
  complete solution to a problem from this
  sectiondo one and submit it or do more for
  extra fun. (Downtown Deli, The Conspirators
  Code, Letter from College)
• Email your complete solution, with your steps
  clearly shown to NCAMath_at_district87.org or put
  them in Nancy Powells mail box!
• Have fun and Problem solve!