Lucas Sequences
A Lucas Sequence is made from two starting terms and a recursive formula for the next term using the two previous ones.
Divisible Sums
Problem of the Week #77: Monday January 20th, 2025As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability…
Friendly Factor Pairs
The number 6 has two pairs of factors: 2x3 and 1x6. These two pairs of factors are called friendly because the sum of one pair is equal to the difference of the other pair, ie. 2+3 = 6-1 = 5.
Digit Sum Divisors
Consider a natural number N and the sum of its digits S. Let’s call a divisor of S a Digit Sum Divisor of N.
Crossing Numbers
From the list of numbers 1 to n, cross out every other number starting with the smallest and repeat the process with the remaining numbers until only one number is left. What number would that be for a given n?
Separating Numbers
Here’s one way to put a pair of 1s, a pair of 2s, and a pair of 3s in a row such that the pair of 1s have one number between them, the pair of 2s have two numbers between them, and the pair of 3s have three numbers between them:
Mod Cycle
Starting with 3, let’s double it and add one (or 2n+1), then modulate the answer by 14 (divide by 14 and take the remainder). If we repeat these two steps, we get the first cycle.
Splitting Ladder
Starting with a natural number at the top, we can split it at each level such that the two lower neighbours always add up to the higher neighbour. Let’s repeat this process until a row can’t be split using whole numbers in the following row anymore.
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.