Getting Started with a Problem

1. 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.”

2. 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… ) http://en.wikipedia.org/wiki/Stirling%27s_approximation

3. 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

4. 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

5. 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

6. 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

7. 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) B A C C B A

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

9. Engagement Example • Cryptoarithmetic problem A D + D I --------- D I D Need to start somewhere, find a chink in the problem’s armor

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

11. 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 3 coins:

12. 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. The solution for 9 coins: More: The 12 coins puzzle the 9 coin interactive

13. 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: Problem 1: It takes one full liliputian glass of their good and strong wine for the liliputian to start feeling joyous. How many such glass does Gulliver need for the same effect? Problem 2: How many liliputian suites needs to be cut and and re-stitched to make a suit to fit Gulliver? Problem 3: According to Gulliver’s story, the giants could walk, run and jumpjust like Gulliver. Why is this not really possible?

14. f4 f2 f3 f1 The problem: Prove that even with the best of today’s computers the protein folding problem can not be solved by an algorithm that performs an exhaustive search for the minimum energy configuration of the polypeptide chain. Ok, now a real real one. Use ANY source of information available, including the web. In fact, you have to, to solve it. Start by figuring out what is being asked. Read about it a bit. Here is what a polypeptide chain looks like. Angles between links can take on any value from 0 to 360 degrees. The set of angles { _k } uniquely specifies the chain. In a typical protein there are k=100 or more links. The biologically functional folded state of the chain ( protein ) has the absolute lowestenergy of all possible configurations. Any two configurations whose angles differby less that 36 degrees (error margin) can be considered identical. 

15. f4 f2 f3 f1 Solution Steps: Google for unknown terminology. Read about the “protein folding” problem. Estimate the number of conformations that you would need to go through (exhaustive search - all of them). A typical protein is a chain of ~ 100 mino acids. Assume that each amino acid can take up only 10 conformations (vast underestimation) Total number of possible conformations: 10100 Suppose each energy estimate is just 1 float point operation. Supposeyou have a Penta-Flop supercomputer (find out what “penta-flop” is). An exhaustive search for the global minimum would take 1085 seconds ~ 3*1078 years. Age of the Universe ~ 2*1010 years.

16. Well, an exhaustive search is really bad. How about something more clever? Tons of methods exist for minimization of a multi-dimensional functions. But, in the case of protein folding we have 1000-dimensional space and very rugged landscape. Lost of local minima. Need just one global one. Very hard.

17. 2 3 Free energy 1 Finding a global minimum in a multidimensional case is easy only when the landscape is smooth. No matter where you start (1, 2 or 3), you quickly end up at the bottom -- the Native (N), functional state of the protein. Folding coordinate Adopted from Ken Dill’s web site at UCSF

18. Realistic landscapes are much more complex, with multiple local minima – folding traps. Proteins “trapped” in those minima may lead to disease, such as Altzheimer’s Adopted from Ken Dill’s web site at UCSF

19. Adopted from Dobson, NATURE 426, 884 2003

20. Intrigued? Suggested reading: 1.“Protein Folding and Misfolding”, C. Dobson, Nature 426, 884 (2003). 2. “Design of a Novel Globular Protein Fold with Atomic-level Accuracy, Kuhlman et al. , Science, 302, 1364, (2003) + references therein.