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Introduction
• Problem avoidance
• Just do it!
• Outlook determines outcome
• The weakest link
• The moving target
• Strategies
• Divide and conquer
• Find a better tool
• Identify a more general problem
• Compute or simplify
• Find a better perspective
• Find a better perspective II
• Find a pattern
• Think in a new dimension
• The greedy method
• Question everything
• Make a model
• Exercise both halves of your brain
• Practice "free association"
• Pending
• Bring me a rock
• Start stupid and evolve
• Stepwise decomposition
• "Exploratory programming"
• Keep it simple
• The 2 rules of reuse

# Outlook determines outcome

### A poet walks through a field and picks up burrs on his wool trousers. He describes the experience esthetically, "How nature attaches itself to man." The same thing happens to a comedian. That night, he gets onstage and complains, "What is it with these burrs? God's way of getting even with me for walking on the grass?" An engineer walks through the field, and then spends 5 years in the lab to invent Velcro.

"Genius is 10% inspiration and 90% perspiration"
Problem solving is 10% reality and 90% mentality
All three experienced the same reality; yet only one saw an opportunity, and solved a problem.

An engineer takes a problem and fixes it.   A humanist takes a problem and celebrates its complexity.
[Robert Weisbuch]

# The weakest link

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# A moving target

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# Divide and Conquer

### Suppose you wrap the earth snugly with a metal band around its circumference (the label C below represents the earth's 25,000 mile circumference). Then you add 10 feet, and distribute the slack evenly (C + 10). How much of a gap has been introduced (the distance X)? How would you solve this problem? What does X represent? (Click on the diagram.) So this problem can be divided into 2 sub-problems: compute R1, and compute R2. We know the formula for the circumference of a circle. We know the value of C. We know how to convert 25,000 miles into a comparable number of feet. How would you compute the value of R1? (Click on the diagram.)   Do the math. (Click)   How would you compute the value of R2? (Click)   Do the math. (Click) Now, X can be computed from the 2 sub-problems. (Click)

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# Find a better tool

### All this math is fairly tedious: very large numbers, lots of zeros, potential round-off error. Is it possible to approach this problem with a little insight, and indulge our natural instinct to be lazy?   If we collected all the equations that describe this problem, could we defer all the arithmetic, and exercise a little leverage with algebra? (Click) Plug equations 2 and 3 into equation 1. (Click)   Expand the expression on the right side. (Click)   Eliminate the factor common to both sides. (Click)   Divide both sides by 2. (Click)   Solve for X. (Click) What happened to R1 and R2?   They are gone!!!   Using this more elegant approach (and resisting the urge to simply "plug and chug") not only eliminated most of the tedious (and error-prone) arithmetic; but it also generated remarkable insight — this problem is totally independent of the size of the sphere!!!

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# Identify a more general problem

### "The more general problem is often easier to solve, and in programming, the more general design may be easier to implement, may operate faster on the machine, and can actually be better understood by the user. In addition, it will almost certainly be easier to maintain, for often we won't have to change anything at all to meet changing circumstances that fall within its range of generality." The previous problem demonstrated 2 different approaches: Focus on the trees, keep your head down, and don't look up until the problem is vanquished. Focus on the forest, and identify a more general problem.

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# Compute or Simplify

### If you drill a 6 inch tall hole through a sphere, what volume of the sphere remains? Click on the diagram, for another representation of the problem.   Do you know the radius of the cylinder?   How about the radius of the sphere?   Using the "more general problem" strategy, the volume of the sphere remaining can be expressed as: volume remaining = (volume of sphere) - (volume of cylinder) - (volume of top and bottom end caps) Using geometry as a tool, the right side results in the following equation: (4/3 * pi * R23) - (pi * R12 * H) - 2 * (1/6 * pi * K * (3 * R12 + K2)) After a page of algebra, we could derive an answer of  –  1/6 * pi * H3 BUT, is there any way to "massage" this problem in order to simplify it?   How many different sized spheres can "contain" a 6 inch tall hole?   How large is the radius of the cylinder?   Do we know?   Do we care?   Click on the diagram to see 4 possible configurations. Is there an inference you can draw from this series of examples that makes the problem trivial?   Click again for the answer.

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# Find a better perspective

## The larger square has an area of 16 square inches. What is the area of the smaller square? Does it help to rotate your perspective?

To rotate, the inner square, click on the diagram.

## Does it help to add some lines?

To add two diagonal lines, click on the diagram.

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# Find a better perspective II

## Can you move two sticks and form the word "cat"? What if you rotated the pattern 90, 180, or 270 degrees?

Click on the diagram four times.

## How about this orientation?   Can you make a 'T' out of the '1'?

Click on the diagram. Click again to see the final answer.

## Is this the only answer?   What about spelling "cat" vertically?

Click on the diagram.

## Can you make a 'T' out of the '1'?

Click on the diagram. Click again to see the final answer.

# Find a pattern

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## Wordle examples: "you just me" encodes the phrase "just between you and me" "COLOWME" encodes the phrase "low income"

For a series of hints, click on the wordle once, then again, then again, ...

# Think in a new dimension

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## What does "move" mean?   Does it mean left or right?   Could it mean up or down?   Are there different ways to represent numbers?

For the answer, click on the statement.

# The greedy method

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# Question everything

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### Did the puzzle statement specify the number 8, or the digit 8?   Is there a difference?   Do we care?

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# Make a model

### Models can help the user to visualize, manipulate, and understand a problem. Modeling is about organizing information, and organization can equip the mind to deal with increasing quantity and complexity of problems. Models allow facts to be separated, sorted, compared, and correlated. The "sorting" allows an overwhelming amount of information to be compartmentalized, and dealt with by our finite minds. The game of Set started as a model for examining epilepsy in German Shepherds. The population geneticist studying the problem used several shapes, colors, and other graphical attributes to model blocks of common and distinct genetic traits. This coded information was recorded on dozens of index cards and spread out on a table. The cards were moved, swapped, and grouped until useful patterns could in discerned. Each card captures 4 dimensions of information: shape (X, O, or rectangle), number (one, two, or three), color (red, green, or blue), and fill (open, striped, or solid). A set consists of 3 cards where each dimension across all 3 cards is the same or different. "Same" means all 3 cards use the same symbol, the same number of symbols, the same color, or the same fill. "Different" means one card is red, one is green, and one is blue; OR, one is open, one is striped, and one is solid; etc. It does not mean one is red, and two are green. Click on the game seven times to see the sets highlighted one at a time.

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# Practice "free association"

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