Notes – Eliminating Possibilities
Eliminating possibilities is a powerful problem solving strategy. It is somewhat of
an indirect method. In other words, we don’t immediately go directly for the
solution, but determine what it _______________________________________.
We use _________________________ in the same manner as Sherlock Holmes did to solve
mysteries. In the book Sign of Four Sherlock Holmes says: “… once you have
eliminated the _________________________________, then whatever is left, no matter how
improbable, must be the ___________________________________.”
Even when we cannot figure out how to get to the exact answer,
_______________________________________________ can help us make a better, educated guess.
Eliminating Possibilities Example
Let’s say that we were playing a 20-questions type math game where I could answer
only “yes” or “no” to any question you asked. In this game I have picked a number
between 1 and 100. How could we use the strategy of eliminating possibilities
along with logic to determine the value of the number?
Each question you ask should be designed to eliminate possibilities. Which of the
following would be the most effective first question?
1. Is the number 29?
2. Is the number even?
3. Is the number greater than 50?
Between 1 and 100 there are _____________ possible whole numbers. Question 1
eliminates one possibility. You have a ___________% chance of guessing correctly, but
this will not be an effective or efficient approach.
Questions 2 and 3 both eliminate ________________% of the possible numbers. These
are _____________________first questions. Combined, they will eliminate _____________% of
the possibilities.
Notice that questions 2 and 3 have __________________% chance of giving you the actual
solution but they help you focus your search. That is the effectiveness and power of
eliminating possibilities.
Example Problem – Penny’s Dimes Part II
Penny’s favorite coin is the dime, as we saw when we were studying systematic lists.
Since we worked the problem about Penny and her dimes, she has spent some of her
dimes and acquired some more. She did not keep track of how many she has. She
knows that it less than 100.
One day she was arranging them in piles on her desk in different ways. The results
were peculiar.
1.
2.
3.
4.
When she put them into piles of 2 there was 1 left over.
When she put them into piles of 3 there was 1 left over.
When she put them in piles of 4 there was 1 left over.
When she put them in piles of 5 there were none left over.
How many dimes does Penny have?
Try This Problem on Your Own and Return to the Video When Complete. Use
the space below. Use additional paper if necessary.
Solution:
Penny has __________dimes
Perhaps you used the ideas below as part of your logic:
1. The clue within the problem statement says that she has less than 100 dimes,
so we need not consider any number above ______________.
2. We need not consider _____________________ values since Penny cannot have
negative dimes.
3. We need not consider ___________________ because it says she was stacking
dimes. __________________dimes cannot be stacked.
Final Thoughts
Eliminating possibilities falls into the category of ____________________ information.
After organizing the information given in a problem, you can eliminate
possibilities. Again, to solve a problem, _____________________ strategies are often
used.
You use the eliminating possibilities strategy in your every day life. For example,
when you decide what you want for breakfast, what to do after school or what
you will write about for an essay you must eliminate possibilities.