Magic, Math, and Infinity

Magic Trick with Cards

• Initial Setup:

  • Start with 27 cards.
  • Arrange them into three columns of nine cards each.

• The Trick:

  • A volunteer selects a card without revealing it.
  • The volunteer identifies the column containing their card (left, middle, or right).
  • The magician collects the cards, placing the selected column in the middle of the deck.
  • This process is repeated three times.
  • The magician then reveals the selected card by spelling out "OUR MAGIC IS REAL," with each letter corresponding to a card.

Explanation of the Card Trick

• Core Principle:

  • The trick relies on ensuring the selected card ends up in the 14th position in the deck (when the deck asis 27 cards).
  • The phrase "OUR MAGIC IS REAL" has 14 letters.

• Steps Breakdown:

  • Taking 27 cards: This sets up the condition for easy middle calculation.
  • Arranging the Cards and Identifying the Column: Gathers necessary information.
  • Placing the Identified Column in the Middle: Moves the range of possible card locations to the center.
  • By the third repetition, the card's position is fixed at the 14th spot.

Advanced Card Maneuvering

• Extending the Trick:

  • After the initial three column repetitions, the magician can manipulate the deck so the selected card ends up at a chosen position.

• Method:

  • The volunteer chooses a number between 1 and 27.
  • A table is used to determine the order in which the columns should be picked up (bottom, middle, top) to move the card to the specified position.

• Ternary Number System:

  • The order (bottom, middle, top) can be determined using the ternary number system (base-3).
  • Top = 0, Middle = 1, Bottom = 2
  • Convert the chosen number (minus one) into its ternary representation to get the pickup sequence.
  • For instance if the number chosen is 20:
  • $20 - 1 = 19$
  • $19 = (2 * 9) + (0 * 3) + (1 * 1)$, or 201 in ternary.

Number Systems

• Common Number Systems:

  • Decimal (base-10): Digits 0-9
    • Example: $237 = (2 * 10^2 ) + (3 * 10^1) + (7 * 10^0)$
  • Binary (base-2): Digits 0 and 1
  • Ternary (base-3): Digits 0, 1, and 2
  • Babylonian cuneiform numbers were base-60

• Ternary Advantages:

  • Historically used in early computers.
  • In some respects constitutes the best system.

• Converting Decimal to Ternary:

  • $20<em>{10} = 202</em>3 = (2 * 3^2) + (0 * 3^1) + (2 * 3^0) = 18 + 0 + 2$
  • To get the card trick to work you must subtract 1 before converting to ternary, so converting 19 to ternary yields 201 as show above.

Cardinality of Infinity

• Types of Infinity:

  • Counting infinity (aleph-zero): The number of natural numbers (1, 2, 3, …).
  • The continuum: The number of real numbers (the number of points on a line).

• Interval and Line:

  • A small interval on a line has the same cardinality as the entire line.
  • This can be shown using stereographic projection.
    • Bend the interval into a semicircle.
    • Project points from the center of the semicircle onto the line, creating a one-to-one correspondence

• Stereographic Projection:

  • Can be used to project circles on a sphere onto a plane.
    • Circles at arbitrary angles on a sphere can be projected into perfect circles.

• Interval and Square:

  • The interval from 0 to 1 has the same cardinality as a square.

• Zip Up Proof:

  • Take a point in the square with coordinates $A = 0.a<em>1 a</em>2 a<em>3 a</em>4 a<em>5 …$ and $B = 0.b</em>1 b<em>2 b</em>3 b<em>4 b</em>5 …$
  • Create a number on the interval from 0 to 1 by interleaving the digits: $0.a<em>1b</em>1a<em>2b</em>2a<em>3b</em>3…$

Infinite Hierarchy of Infinities

• Power Set:

  • The power set of a set A is the set of all subsets of A.
  • The cardinality of the power set of A is always greater than the cardinality of A.
  • |A| < |P(A)|

• Example:

  • Let $A = {1, 2, 3}$
  • Then $P(A) = { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }$

• Constructing Larger Infinities Iteratively:

  • Start with the natural numbers $\mathbb{N}$.
  • Take the power set of $\mathbb{N}$, then take the power set of that, and so on.
  • Each time, a larger infinity is created.

• Cardinal's Diagonal Argument:

  • How to prove that the power set of A is always bigger than A
  • Assume that there is a function $f$ that maps elements of A to subsets of A (i.e., elements of P(A)).
  • Construct a set $B$ as follows: An element $x$ of A is in $B$ if and only if $x$ is not an element of the subset that $f$ maps $x$ to.
    • In set notation:
      • $B = { x \in A \mid x \notin f(x) }$
  • Now, the critical part: because $B$ is a subset of $A$, it must be in the power set $P(A)$.
    • If the mapping is onto, then there must be an element $y$ in $A$ such that $f(y) = B$.
    • Two possibilities:
      • Suppose that $y$ is in $B$. By the definition of $B$, this means $y$ is not in $f(y)$. But $f(y)$ is $B$, so $y$ is not in $B$. Contradiction.
      • Suppose that $y$ is not in $B$. Then $y$ is not in $f(y)$. By the definition of $B$, this shows that $y$ must be in $B$. Contradiction.
    • Both possibilities lead to contradictions, so our original assumption that the mapping from $A$ to $P(A)$ is onto leads to a contradiction. So there can be no onto mapping, which means that $P(A)$ must be bigger than $A$.