Slide1: Eliminate Possibilities Problem Solving Strategy


Eliminate Possibilities: 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.


Eliminate Possibilities: 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. You’ll need to keep reviewing the clues with each case. Here is a diagram that illustrates the thinking process used in this strategy. Assumption doesn’t work! It is wrong – eliminate it! Make assumption Apply the assumption in the problem Assumption works! We didn’t learn much – it’s still a possibility!


Eliminate Possibilities: Eliminate Possibilities You probably apply this strategy when you go to a restaurant to eat. You eliminate what you don’t 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.


Cryptarithmetic Problems: 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 problem…A 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!


Slide6: Olive: Hey, that sounds like a good cryptarithmetic problem. Let’s 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. Emil: Gordan’s tomcat and Pearl’s old nag were not much use. I traded them for my new goat. Cryptarithmetic Problem:


Make systematic lists.: Make systematic lists. G T O M P N A E 5 0 1 2 3 4 5 6 7 8 9 G


Generate possibilities.: Generate possibilities. In the thousands column, 5 + P adds to 5. So P = 0 and there’s no carry to this column or P = 9 and there’s a carry of 1 to this column.


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. The thousands column adds to two digits, so P is not 0. P = 0 or P = 9


List the remaining possibility.: List the remaining possibility. Therefore, P = 9, there’s a carry to this column, and E = 1. 0 1 2 3 4 5 6 7 8 9 E G P G T O M P N A E 5 9 1


Generate possibilities.: In the tens column, O + A adds to A. So O = 0 and there’s no carry to this column or O = 9 and there’s a carry of 1 to this column. Generate possibilities.


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. P = 9, so O ≠ 9. O = 0 or O = 9 0 1 2 3 4 5 6 7 8 9 E G P G T O M P N A E 5 9 1


List the remaining possibility.: List the remaining possibility. Therefore O = 0 (zero). G T O M P N A E 5 0 9 1 0 1 2 3 4 5 6 7 8 9 O E G P


Generate possibilities.: Generate possibilities. In the hundreds column, T + N = 10. So we have these possibilities for T and N:


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. G T O M P N A E 5 0 9 1 0 1 2 3 4 5 6 7 8 9 O E G P T and N can’t be 1, 5, or 9 because these numbers are already used.


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. T can’t be 2, 3, or 4 because M + 5 = T in the ones column, and M can’t be negative. 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


List the remaining possibilities.: 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


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. M can’t be 1 because E = 1. 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


Reread the problem.: 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.


Eliminate possibilities by finding contradictions.: Eliminate possibilities by finding contradictions. M can’t be 2 because then T and N would be the unused odds, leaving none for A. 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


List the remaining possibility.: List the remaining possibility. T 8 N 2 M 3 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


Finish the problem and check.: 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


Strategy: Eliminate possibilities: 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.


Now it’s your turn…: Now it’s 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 section…do 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@district87.org or put them in Nancy Powell’s mail box! Have fun and Problem solve!