THE A LEVER FOR MIND @tomstuart at RubyConf on 2014-11-19

Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” — Archimedes “

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ABSTRACTION

NUMBERS

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3

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4 3 2

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<

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3

2 3 5 combine add count count count REAL ABSTRACT

2 3 5 add ABSTRACT

SUMS

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ?

1 + 2 = 3 3 + 3 = 6 6 + 4 = 10 10 + 5 = 15

15 + 6 = 21 21 + 7 = 28 28 + 8 = 36 36 + 9 = 45 45 + 10 = 55

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

1 + 2 + 3 + 4 + 5 + … … + 98 + 99 + 100 = ?

? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

? 1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 =

11 + 11 + 11 + 11 + 11 = ?

11 + 11 + 11 + 11 + 11 = 55

1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 = 55

1 + 2 + … + 9 + 10 = (10 ÷ 2) × (10 + 1)

1 + 2 + 3 + 4 + 5 + … … + 98 + 99 + 100 = ?

1 + 100 + 2 + 99 + 49 + 52 + 50 + 51 = ? ⋮

101 + 101 + 101 + 101 = ? ⋮

101 + 101 + 101 + 101 = 5,050 ⋮

1 + 2 + … + 99 + 100 = (100 ÷ 2) × (100 + 1)

1 + 2 + … + n = (n ÷ 2) × (n + 1)

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10 11

(n + 1) × n ÷ 2 1 + 2 + … + n =

1 + 2 + … + n = (n + 1) × n ÷ 2 ( )

1 + 2 + … + 999 + 1000 = ?

1 + 2 + … + 999 + 1000 = (1000 ÷ 2) × (1000 + 1)

1 + 2 + … + 999 + 1000 = 500 × 1001

1 + 2 + … + 999 + 1000 = 500,500

GRAPHS

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5 3 3 3

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DEGREE 2

DEGREE 3

WALK

CONNECTED

CONNECTED

CONNECTED

CONNECTED

TRAIL

EULERIAN TRAIL

EULERIAN TRAIL CONNECTED OF ODD DEGREE AT MOST 2 NODES

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TRIANGLES

4 3 5

4 4 5.7

4 5 6.4

3 5.8 5

a b c a² + b² = c²

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c²

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a² b²

c² a² b²

c² a² b² + =

c² a² b²

c² a² b²

c² a² b²

a b c a² + b² = c²

MATHEMATICS

SPOT PATTERNS, BUILD REUSABLE ABSTRACTIONS

BAD MATHEMATICS EDUCATION

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POKER

HIGH CARD ONE PAIR TWO PAIR

THREE OF A KIND STRAIGHT FLUSH

Card = Struct.new(:rank, :suit) three_of_diamonds = Card.new(3, :diamonds) queen_of_hearts = Card.new(12, :hearts) ace_of_spades = Card.new(14, :spades)

Hand = Struct.new(:cards) do include Comparable def <=>(other) # TODO end end

Hand = Struct.new(:cards) do include Comparable def <=>(other) highest_card_rank <=> other.highest_card_rank end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if empty? && other.empty? 0 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if one_pair? if other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if two_pair? if other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if three_of_a_kind? if other.three_of_a_kind? if highest_triple_rank == other.highest_triple_rank without { |c| c.rank == highest_triple_rank } <=> other.without { |c| c.rank == highest_triple_rank } else highest_triple_rank <=> other.highest_triple_rank end else 1 end elsif two_pair? if other.three_of_a_kind? -1 elsif other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? || other.three_of_a_kind? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? || other.three_of_a_kind? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end

Hand = Struct.new(:cards) do include Comparable def <=>(other) if straight? if other.straight? if highest_straight_rank == other.highest_straight_rank without { |c| c.rank == highest_straight_rank } <=> other.without { |c| c.rank == highest_straight_rank } else highest_straight_rank <=> other.highest_straight_rank end else 1 end elsif three_of_a_kind? if other.straight? -1 elsif other.three_of_a_kind? if highest_triple_rank == other.highest_triple_rank without { |c| c.rank == highest_triple_rank } <=> other.without { |c| c.rank == highest_triple_rank } else highest_triple_rank <=> other.highest_triple_rank end else 1 end elsif two_pair? if other.three_of_a_kind? || other.straight? -1 elsif other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? || other.three_of_a_kind? || other.straight? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? || other.three_of_a_kind? || other.straight? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end

Card = Struct.new(:rank, :suit) class Card::Rank TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } end class Card::Suit CLUBS, HEARTS, SPADES, DIAMONDS = 4.times.map { new } end three_of_diamonds = Card.new(Card::Rank::THREE, Card::Suit::DIAMONDS) queen_of_hearts = Card.new(Card::Rank::QUEEN, Card::Suit::HEARTS) ace_of_spades = Card.new(Card::Rank::ACE, Card::Suit::SPADES)

class Card::Rank include Comparable ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def <=>(other) ORDER.index(self) <=> ORDER.index(other) end end

class Hand::Category include Comparable ORDER = ( HIGH_CARD, ONE_PAIR, TWO_PAIR, THREE_OF_A_KIND, STRAIGHT, FLUSH, FULL_HOUSE, FOUR_OF_A_KIND, STRAIGHT_FLUSH = 9.times.map { new } ) def <=>(other) ORDER.index(self) <=> ORDER.index(other) end end

class << Hand::Category::HIGH_CARD def matches?(hand) true end def compare(a, b) # … end end

class << Hand::Category::ONE_PAIR def matches?(hand) pairs(hand).size == 1 end def compare(a, b) # … end end

Hand::Rank = Struct.new(:hand) do include Comparable def category Hand::Category::ORDER. select { |category| category.matches?(hand) }.max end def <=>(other) if category == other.category category.compare(hand, other.hand) else category <=> other.category end end end

Hand = Struct.new(:cards) do def rank Hand::Rank.new(self) end end if my_hand.rank > your_hand.rank puts 'I win!' end

class Card::Rank ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def succ ORDER[ORDER.index(self).succ] end end

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low high

class Card::Rank ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def succ ORDER[ORDER.index(self).succ % ORDER.size] end end

“DESIGN”?

When a thing looks complicated, it’s possible that we’re looking at it wrong and missing some of the pieces of the puzzle.” — Richard Feynman “

HOW?

FIND THE HONEST ABSTRACTION

FIND THE MINIMAL ABSTRACTION

FIND THE INTERESTING THING INSIDE THE BORING THING

BE EXCITED!

KEEP YOUR PERSPECTIVE

IF YOU CAN WRITE A PROGRAM, YOU’RE GOOD AT MATHEMATICS.

SORRY!* * not sorry

THANKS @tomstuart, [email protected], http:/ /codon.com/