Getting Started with a Problem “Eighty percent of success is showing up.” –Woody Allen “Success is 1% inspiration and 99% perspiration.” – Thomas Edison. To successfully solve any problem, the most important step is to get actively involved. –The Principle of Intimate Engagement: You must commit to the problem –“Roll up your sleeves” –“Get your hands dirty.” 1

Easy vs. Hard Problems Exercises: (e.g. compute 10! without a calculator) Easy problems: See the answer Medium problems: See the answer once you engage Hard problems: You need strategies for coming up with a potential solution, sometimes for even getting started. Open-ended problems are often like this. Often, multiple possible solutions, you need a strategy to choose “the best” one. (e.g. Estimate 2710! Why is this one hard? You can use a calculator… ) 2

Effective vs. Ineffective Problem Solvers Effective: Believe that problems can be solved through the use of heuristics and careful persistent analysis Ineffective: Believe ``You either know it or you don't.'' Effective: Active in the problem-solving process: draw figures, make sketches, ask questions of themselves and others. Ineffective: Don't seem to understand the level of personal effort needed to solve the problem. Effective: Take great care to understand all the facts and relationships accurately. Ineffective: Make judgments without checking for accuracy 3

Mental Toughness Need the attributes of confidence and concentration –Confidence comes with practice –Attack a new problem with an optimistic attitude Unfortunately, it takes time –You can’t turn it on and off at will –Need to develop a life-long habit 4

Engagers vs. Dismissers Engagers typically have a history of success with problem solving. Dismissers have a history of failure. You might be an engager for one type of problem, and a dismisser for another. You can “intervene with yourself” to change your attitude of dismissal 5

The Mental Block Many students do significant problem solving for recreation –Sodoku, computer games, recreational puzzles. These same students might dismiss math and analytical computer science problems due to a historical lack of success (the mental block) And the other way around: dismissing verbal problems. To be successful in life you will need to find ways to get over any mental blocks you have Learn to transfer successful problem-solving strategies from one part of your life to other parts. –Example: Writing is a lot like programming 6

Example Problem Connect each box with its same-letter mate without letting the lines cross or leaving the large box. (Actual problem used in software company job interview) A B C CB A 7

Strategy (my favorite): solve a simpler problem first. A B C CB A 8 CA

Heuristic: Wishful Thinking A B C CB A C B A CB A A B C CB A C B A CB A

Engagement Example Cryptoarithmetic problem (base 10) A D + D I D I D 10 Need to start somewhere, find a chink in the problem’s armor

The 9 coin problem 9 coins that look alike. One is fake, can be heavier or lighter (not known). Using a simple balance scale and 3 weighing, single out the fake one. Hint: solve a simpler problem first. Which one?

The solution for 3 coins: the weighings are: 1 against 2 1 against 3 Both of these can have three outcomes: fall to the left (l), fall to the right (r), or balance (b). The following table gives the answer for each of these outcomes: outcome fake coin # why: l l 1 too heavy l b 2 too light l r (not possible) b l 3 too light b b no fake coin b r 3 too heavy r l (not possible) r b 2 too heavy r r 1 too light

The solution for 9 coins: Step 1. Divide 9 coins into 3 piles of 3 coins each. Use the 3-coin strategy to weigh: pile 1 against pile 2 pile 1 against pile 3 From Step 1, you will determine: a) which pile contains the fake and b) if the fake is heavier or lighter. Step 2 : Weigh 2 coins from the pile that contains fake. Total # of weighings: 2+1 = 3. More: The 12 coins puzzleThe 12 coins puzzle the 9 coin interactive

Let’s get closer to the real world: In “Gulliver’s Travels”, by Swift, Gulliver travels to distant parts of the Earth where he meets liliputians who are mere 6 inch tall, and giants who are 72 feet tall. These people are made of the same kind of flesh, muscle and bone; and look pretty much like like Gulliver who is 6 feet tall.

Let’s get closer to the real world: In “Gulliver’s Travels”, by Swift, Gulliver travels to distant parts of the Earth where he meets liliputians who are mere 6 inch tall, and giants who are 72 feet tall. These people are made of the same kind of flesh, muscle and bone; and look pretty much like like Gulliver who is 6 feet tall. Problem 1: It takes one full liliputian glass of their good and strong wine for the liliputian to start feeling joyous. How many such glasses does Gulliver need for the same effect? Problem 2: How many liliputian suites needs to be cut and re-stitched to make a suit to fit Gulliver? Problem 3: According to Gulliver’s story, the giants could walk, run and jump just like Gulliver. Why is this not really possible?

Key steps: Problem 1. The larger the volume of the person, the more wine. Number of glasses ~ volume. Need to find ratio of volumes Gulliver/Lilliputian. Key simplification for problems 1-2. Assume human to be a cube of side a. Then V = a 3. When a increased by 12, V=a 3 goes up by For #2, we need surface area, SA = 6a 2. Gulliver’s area ~ 12 2 x SA(liliputian). For problem #3 needs a more sophisticated simplification. Now, the mass of the body that the bone/leg/muscle has to support is proportional to size a. While the strength of the bone/muscle is proportional to its cross-sections, that is a 2. As a grows, the ratio of Mass to strength becomes smaller. So, the giants will crush their bones if they tried to stand up. By the same argument, an ant can lift 40 times its own weight, while a human can only lift about as much as his own weight only. 16 a Leg Body

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