Covering Squares
We choose one space in a rectangular grid made of unit squares, and we count the number of squares or rectangles that cover this space.
Mathematics
Mirror Functions
Let’s consider all real valued functions whose inverse is the function itself. Since the graphs of these functions are their own mirror images across the line y=x, let’s define these functions as mirror functions.
Counting Rectangles Extended
Continuing from last week’s problem about counting squares, let’s look at the related problem of counting the number of rectangles in a grid, with extensions
Counting Squares Extended
Consider the widely known problem of counting the number of squares in a square grid. Today we will try to extend the problem.
Water Jugs
Consider this family of popular puzzles: You have two empty water jugs with no markings. One holds exactly 8 liters of water when full, and the other holds 3 liters. You also have a full tub with an endless supply of water and another large unmarked empty jug of unknown volume. The goal is to measure out a given volume of water to put into the large jug using the other two jugs.
Curve Stitching
Starting with a square, choose two points on the perimeter of the square to make a line, and then move both end points incrementally in the same direction (clockwise or counterclockwise) and the same distance along the perimeter of the square to make subsequent lines.
Peg Solitaire
In a game of Peg Solitaire, 32 pegs are arranged in a cross shaped grid with the center peg removed. In each move, the player is allowed to move one peg over another, vertically or horizontally onto an empty space, and remove the peg that was jumped over. The game continues until no legal move is available, and the goal is to have as few pegs remaining as possible when the game ends. A perfect game would end with a single peg back in the center of the board.
Tower of Hanoi
Tower of Hanoi is a one-player game using three vertical rods and a number of disks, all of different diameters. The goal of the game is to move the entire pile of disks onto one of the other two pegs.
Number Bases
The standard number system we use today is called decimal numbers, or base 10, meaning at every digit, we count to 9 before carrying one to the next smallest digit.
Continued Fractions
A continued fraction is a way to represent any real number as a sequence of integers by writing down the integer part of the real number and taking the reciprocal of the remaining part at each iteration.