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{10} = 2023 = (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.a1 a2 a3 a4 a5 … and B = 0.b1 b2 b3 b4 b5 …
    • Create a number on the interval from 0 to 1 by interleaving the digits: 0.a1b1a2b2a3b3…

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.