CSE Blog - quant, math, computer science puzzles

Source: Sent to me by Sudeep Kamath Problem: Airline check-in baggage has size restriction by ​so-called ​linear dimension: length + breadth + height should not exceed 62 inches. Prove that you can't "cheat" by packing a box with higher linear dimension into a box with ​lower​ linear dimension. Solution: Posted by GoKu in comments!

Source: Mailed to me by Kushagra Singhal, Ex-IIT Kanpur, PhD Student at University of Illinois at Urbana-Champaign Problem: Prove that for any positive K and a natural number n, every (n*K)th number in the Fibonacci sequence is a multiple of the Kth number in the Fibonacci sequence. More formally, for any natural number n, let F(n) denote Fibonacci number n. That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n). Prove that for any positive K and natural n, F(n*K) is a multiple of F(K). Solution: Posted by Alex_IITD in comments!

Source: Mailed to me by JDGM ("regular commenter JDGM") Problem: The integers greater than zero are painted such that: • every number is either gold or silver. • both paints are used. • silver number + gold number = silver number • silver number * gold number = gold number Given only this information, for each of the following decide whether it is a gold number, a silver number, or could be either: 1.) gold number * gold number 2.) gold number + gold number 3.) silver number * silver number 4.) silver number + silver number Solution: Highlight the part between the * symbols for the answer. * First, the whole colouring is forced into one shape: pick any G > 1 and let gold = multiples of G, silver = everything else. Proof: 1 is silver (if 1 were gold, then silver x 1 would have to be gold by rule 2 - contradiction). Let G be the smallest gold number, so 1..G-1 are silver. Rule 1 (silver + gold = silver) then makes every non-multiple of G silver (write it as kG + r, 0 < r...