Some fractions look like they simplify by incorrectly cancelling digits — yet give the correct result anyway. Take 49/98. Cancel the 9s and you get 4/8 = 1/2. And 49/98 = 1/2. ✅ These are called digit-cancelling fractions — and there are exactly 4 non-trivial ones where both numerator and denominator are two-digit numbers. Project Euler #33 asks you to find them all, multiply them together, and return the denominator of the result in its lowest terms. My approach in Python: →

Problem: Starting in the top-left corner of a 20×20 grid, and only being able to move right and down, how many routes are there to the bottom-right corner? At first glance, this seems like a classic dynamic programming or pathfinding problem. But I took a different approach - let the computer explore small cases, then use pattern recognition to crack the larger problem. from itertools import permutations import math from math import factorial dictionary_x = {} list_x = [0, 1