Balancing Numbers
Consider the set of natural numbers 1 to 8. For this set of numbers, 6 is called the balance because the numbers before 6 add up to the same sum as the numbers after 6. In other words, 1+2+3+4+5 = 7+8.
Fractions of a Square
The original problem asks to cut a square by drawing straight lines between any number of midpoints of sides and corners of the square such that a resulting region is 1/5 of the area of the whole square.
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.
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.
Perfect Numbers
A perfect number is a natural number that is equal to the sum of all of its proper divisors (divisors that are less than the number itself). For example, 6 is the smallest perfect number because its proper divisors, 1, 2, and 3, add up to exactly 6.
Splitting LCMs (reworded)
Consider all pairs of natural numbers that add to 9, which pair has the largest possible Least Common Multiple?
Triangular and Trapezoidal Numbers
Triangular numbers and trapezoidal numbers are both natural numbers that can be expressed as the sum of more than one consecutive natural numbers.