How do you figure things out? | 78 ways |

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Get your hands dirty * * *So we try another strategy, one of the best for beginning just about any problem: get your hands dirty. We try plugging in some numbers to experiment. If we are lucky, we may see a pattern. ... This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech "plug and chug" methods. The great Carl Gauss ... was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x**2+y**2<=90,000. ... Don't skimp on experimentation! Keep messing around until you think you understand what is going on. Then mess around some more.* pg.7, 30, 36 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1412

Knowing when to give up * * *Sometimes you just cannot solve a problem. You will have to give up, at least temporarily. All good problem solvers will occasionally admit defeat. An important part of the problem solver's art is knowing when to give up.* pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1414

Practice * * *Practice by working on lots and lots and lots of problems. Solving them is not as important. It is very healthy to have several unsolved problems banging around your conscious and unconscious mind.* pg.25, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1423

Toughen up * * *Toughen up by gradually increasing the amount and difficulty of your problem solving work.* pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1424

Space | ||||

Space Axiom of the empty set Euler's formula Change your point of view The swimmer's hat Considering the simplest point in an arbitrary coordinate system Invariant with respect to permutation of some of the roots of a polynomial Notation Add zero creatively Completing the square by adding zero Easy invariants Homogeneous coloration Modular arithmetic Multiply cleverly by one Parity 127 people in a tennis tournament Parity problem: Dominos on a chessboard Parity of a sum or product | ||||

Sequence | ||||

Sequence Axiom of Infinity Standard induction Prove n! > 2**n where integer n >= 3 Prove that the sum of the interior angles of any n-gon is 180(n-2) degrees Extremal arguments involving infinite sets Use of extreme principle with induction Well-Ordering Principle Number of games in an elimination-style tournament Finite Injury Priority Method | ||||

Limits | ||||

Limits Infinite Injury Priority Method Generalizing the scope of a problem Stretching things Topologically equivalent Define a function Geometric series tool Restate Axiom schema of specification Four bugs chasing each other Geometric symmetry Fetching water for Grandma Square inscribed in circle inscribed in square Invariant with respect to transformations Not quite symmetrical Search for order Symmetry | ||||

Truth | ||||

Truth Argument by contradiction Square root of 2 is not rational Deliberately misleading presentation Recast a problem from one domain into another domain Deduction Penultimate step Recast geometry as logic Free variables and Bound variables Bend the rules Draw a picture Drawing the monk problem Loosen up Peripheral vision Without loss of generality | ||||

Restructuring | ||||

Restructuring - 10 Evolution: Hierarchy => Sequence (for determining weights)
- 20 Atlas: Network => Hierarchy (for determining connections)
- 21 Canon: Sequence => Network (for determining priorities)
- 32 Chronicle: Sequence => Hierarchy (for determining solutions)
- 31 Catalog: Hierarchy => Network (for determining redundancies)
- 30 Tour: Network => Sequence (for determining paths)
Models of Multiplication - The addition rule is at work, adding exponents. Multiplying by 10 or dividing by 10 shifts the number with regard to the decimal point, although it looks like the decimal point is moving. We may think of this as simply changing the units, the base unit.
- I think of rescaling as a product of actions that either make bigger (numerator) or make smaller (denominator). They are all multiplying against some unknown, acting upon it. I call the actions "multiplication drops", either "magnifying drops" (say, multiplying by 10) or "shrinking drops" (dividing by 10). So these are actions x actions x (object with units). Thus magnifying and shrinking can cancel out. Also, actions can be decomposed into component actions, into primes.
- Repeated addition is a recounting, a shift from larger units to smaller units. 3 x (23 x dollars) becomes (3 x 23) x dollars. Amount x (large unit) becomes action x (amount x small unit) becomes (action x amount) x small unit becomes amount x small unit.
- Multiplication can give the ways of matching units, multiple units times multiple units, as in box multiplication, accounting for all possibilities. Units times units means that conditions are satisfied, thus generating all of the solutions.
- Multiplication can be thought of as counting items that have been grouped where each group has the same number of items. For example, we can count coins by grouping together the pennies, nickels, dimes, quarter, placing them in rows or groups of 4 or 5 or 10. Number x (Value x Unit).
- Dividing out, for example, money per person. This is like multiple units. (Number of cycles) x (Number of people x Units )
Axiom of pairing Dividing into cases Fractal multiplication: Recopying the whole All parabolas have the same shape Axiom of extensionality Coloring Graph Handshake Lemma Sleeping mathematicians Proportion multiplication: Rescaling the whole Strong induction Tally multiplication: Rescaling the multiple Axiom of power set The two ropes Reimagining the monk problem | ||||

Decomposition | ||||

Decomposition Axiom of union Reorganizing addition to do it in your head Reorganizing multiplication to do it in your head And Binomial theorem Complete the square Conspicuous ceiling Eliminate radicals Extracting squares Factor Theorem Factoring Fundamental Theorem of Arithmetic Halt or repeat Intermediate pigeonhole Number theoretic functions Or Partitions Pigeonhole principle Two points a mile away Repeated application of pigeonhole principle The Factor Tactic x**2 + y**2 = 13 | ||||

Directed graph | ||||

Directed graph Connectivity and Cycles Division Algorithm
Eulerian Path Hamiltonian Path Modulo n filtering Viewing a problem modulo m 10 + 4 = 2 Axiom schema of replacement | ||||

Other ways: Math | ||||

Other ways: Math Alternatives to mathematical induction Epsilon induction Forcing (in set theory) Thought experiments in Mathematics |