Dork Scratchings

    I was given a new version of Dobble for Xmas called Dobble Connect.  Dobble consists of stack of cards with pictures on it where each pair of cards has one match, and each pair of symbols appears on one card.  This is a new version which has 91 symbols in total, 10 symbols per card, and 91 cards.  The fingerprint of the original Dobble was (57 symbols, 8 symbols, 57 cards). An additional innovation in Dobble Connect is that the cards are hexagonal and grouped into a colour per player.  Players build a tiling on the tabletop, placing cards next to each other as soon as they spot a matching symbol between their top card and one of the cards already placed.  The first player to get four of their own colour in a row wins. How is it possible for each pair of cards to have one and only one symbol in common, and for each pair of symbols to appear on one and only one card?  Does that have something to do with the strange number 91?  The answer i...

Rolling marbles  Here's a puzzle: Suppose you have N identical marbles rolling along a one dimensional table-top.  Each marble is randomly rolling to the left or to the right, all with the same speed.  Collisions are elastic, which means the marbles just change direction.  What is the maximum amount of time before all the marbles have rolled off the table? 8 marbles with speed 1 on a table of length 1 Answering this question is really difficult if you simply pick an individual marble and try to work out how long it might stay on the table as it bounces back and forth.  But there's a simpler way to look at it. Prior to each collision you have one marble rolling to the left and one to the right, and afterwards you still have one rolling to the left and one to the right.  If we swap labels following each collision then the labels never change direction.  Now it's easy to see that the answer is the same whether there are 100 marbles or just one.  And ...