Category Archives: Miscellany

Envelopes

There are 13 envelopes, and each one is either red or blue.  There are 5 of one color, each containing $100, and 8 of the other color that are empty.  You don’t know which color is the winning color other than it is the one with the smaller number of envelopes.

At the start of the game, one of the thirteen envelopes appears on your computer screen and stays there for 15 seconds.  During that time interval, you have two options.

(1)   You can pick the envelope and see what’s inside, which ends the game, or

(2)   You can pass on the envelope.  If you pass, the envelope disappears forever and brings up one of the remaining envelopes for you to choose or let pass.

  • How should you play the game to maximize your chance of getting the $100?
  • How would the winning strategy change if the amount of money in each winning envelope decreases by $5 each time you pass on an envelope?

Suppose there are  winning envelopes and  empty envelopes.  How should you adjust your strategy given different numbers of winning and empty envelopes?

Chutes and Ladders

To play Chutes and Ladders, you begin with your marker off the board.  Roll a die and move to the position on the board given by the die.  If you land at the bottom of a ladder, you exit the top.  If you land at the top of a Chute, you slide to the bottom.  To win the game of Chutes and Ladders, you must land exactly on cell 100.   Try 9 and 16 cell games to develop a theory of Chutes and Ladders related to the number of moves required to play the game.   How does the length of the game depend upon the number of sides to the die that is used?  How is a game with no chutes or ladders affected by the number of faces on the die?   How does adding a ladder or a chute affect the length of a game.  Does it depend upon where the ladder is added?

graph1

Reversible Cellular Automata

One-dimensional cellular automata are closely associated with dynamical systems and modern chaos theory. While the results are complex and intriguing, the basic principle of cellular automata is simple:  Given an infinite row of cells with each cell colored either black (on or 1) or white (off or 0).  With each iteration, the cells are recolored based upon the colors of their neighbors.

A simple example of a transition rule is the following:  If the sum of  values of the two adjacent squares is even (equal to 0 (mod 2)), color the cell white in the next iteration, otherwise color it black. Under this rule, the initial row becomes:

graph1

 The simplest way to see the effects of a transition rule is to show the evolution of the row over time.  Thus, each row represents one iteration of the transition rule.  Below we see the first 4 iterations of the transition rule described above.

graph2

Which translates into the following diagram using white and black instead of 0’s and 1’s.

graph3

When describing the cellular automata with a sequence of successive rows, it is helpful to describe the transition rule locally.  The new value of a cell in the \(n^{th}\) row is determined by the values lying directly above it in the \((n-1)^{th}\) row.  Thus:

\(frac{111}{0}  frac{110}{1}  frac{101}{0}  frac{100}{1}  frac{011}{1} frac{010}{0}  frac{001}{1}  frac{000}{0} \)

describes the transition rule given above.  The upper part shows one of the eight possible states that three adjacent cells can have at iteration \(n\), where the lower part shows the state of the central cell of the trio time \(n+1\).  Since \(01011010 = 90\) in base two, the rule above is called the rule of 90.

For which Rules and which size neighborhoods is the cellular automata reversible (that is, you can determine the \((n-1)^{th}\) row from the \(n^{th}\) row?  Does each Rule have an inverse Rule?  Can Rules be composed and if so, does order matter?  Suppose the next state depends on the previous two states, does this create any new structures?

The Prime Derivative

Given a positive integer \(x\), define the derivative of \(x\), denoted, \(x’\) as:

\(x’ = 1\) if \(x\) is a prime.

\(x’={a}\cdot{b’} + {a’}\cdot{b}\) if \(x={a}\cdot{b}\) (\(x\) is composite)

\(x’=0\) if \(x=1\)

Since all composites are products of primes, we can find the derivatives of all orders for all integers.  What patterns are observed?