Problem Solving, Part 1

Troy Vasiga

Centre for Education in Mathematics and ComputingFaculty of Mathematics, University of Waterloo

cemc.uwaterloo.ca

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Outline

• Short biography of the speaker

• Roles and Responsibilities of Math Circles

• Comparing Exercises, Questions and Problems

• Basics of Problem Solving

• Notation

• Symmetry

• Diagrams

• Work on problems

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Exercises, Questions and Problems

Let’s compare and contrast:

• Exercises

• Questions

• Problems

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What makes a good exercise?

You perform exercises to:

• strengthen fundamentals

• solidify concepts

• improve speed

Example Exercise:Factor 221.

Textbooks are full of exercises.

Most exercises should reinforce one small concept.

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What makes a good exercise?

You perform exercises to:

• strengthen fundamentals

• solidify concepts

• improve speed

Example Exercise:Factor 221.

Textbooks are full of exercises.

Most exercises should reinforce one small concept.

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Questions

Similar to scratch-off lottery tickets.

There is some interesting math under the surface, but requires atiny bit of scratching to see it.

Example Question: (2011 Fermat, Question A8)The number halfway between 1

12 and 110 is:

(a) 111

(b) 1120

(c) 1160

(d) 11120

(e) 122

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Qualities of good questions

• Requires a bit more care in the solution

• Requires some setup of equations, unknowns, values

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Qualities of good questions

• Requires a bit more care in the solution

• Requires some setup of equations, unknowns, values

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Problems

Should be like a scavenger hunt.

Or the Lord of the Rings.

Not all those who wander are lost.

–J. R. R. Tolkien

Should involve some analysis, thought, trial-and-error, insight.

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Problems

Should be like a scavenger hunt.

Or the Lord of the Rings.

Not all those who wander are lost.

–J. R. R. Tolkien

Should involve some analysis, thought, trial-and-error, insight.

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Problems

Should be like a scavenger hunt.

Or the Lord of the Rings.

Not all those who wander are lost.

–J. R. R. Tolkien

Should involve some analysis, thought, trial-and-error, insight.

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First steps in problem solving

When solving problems, keep in mind some basic techniques:

• Write down what you know• What am I given?• What am I being asked to solve?

• Communicate your steps and decisions

• Solve the problem being asked

• Stop, drop and look for elegance

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A first problem

John and Mary wrote the Euclid Contest. Two times John’s scorewas 60 more than Mary’s score. Two times Mary’s score was 90more than John’s score. Determine the average of their two scores.

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A second problem

The five expressions 2x + 1, 2x − 3, x + 2, x + 5, and x − 3 can bearranged in a different order so that the sum of the first threeexpressions is 4x + 3 and the sum of the last three expressions is4x + 4. What is the middle expression in this new list?

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Notation

• Quite often, good notation makes problems easier to solve

• Writing good notation makes the solver see the problem in anew light...

• ...or in some familiar light.

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Notation

• Quite often, good notation makes problems easier to solve

• Writing good notation makes the solver see the problem in anew light...

• ...or in some familiar light.

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Notation

What is the probability that n3 ends in 11?

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Notation

You are arranging races for 25 horses on a track that canaccommodate 5 horses at a time. Each horse always runs thedistance in the same time and the horses have distinct speeds. Youhave no stopwatch, but can make deductions from the finishingorder in the races. What is the smallest number of races needed todetermine the 3 fastest horses, in order?

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Symmetry

• Mathematics is a science of patterns

• Lots of patterns have symmetry or similarity

• Very often, spotting symmetry cuts the amount of work by 12 ,

or 14 or more

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Symmetry

Imagine the following game:

• 8× 8 board.

• In one move, a player may move any number of squares to theright or up, but not both.

• The winner is the person who makes it to the top-right corner.

• Player A goes first, B goes next.

• Who wins, and why?

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Symmetry

Imagine the following game:

• n ×m board.

• In one move, a player may move any number of squares to theright or up, but not both.

• The winner is the person who makes it to the top-right corner.

• Player A goes first, B goes next.

• Who wins under which circumstances and why?

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Symmetry and notation

Simplify (a + b + c)(a2 + b2 + c2 − ab − ac − bc).

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Diagrams

• When you are given a problem, draw a picture

• Plot points

• Label edges of diagrams

• Draw a doodle to explain/visualize what the problem is asking

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Diagrams

Determine the area of the triangle with vertices A(1, 3), B(8, 0),C (4, 8).

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Diagrams

In the diagram 4ABC is isosceles with AC = BC = 7. Point D ison AB with ∠CDA = 60◦, AD = 8 and CD = 5. Determine thelength of BD.

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