Here’s a rough summary of February’s London MathsJam. There seemed to be some loose themes, but sadly no pancakes (it was on Shrove Tuesday). Peter Rowlett briefly visited, but left before most people turned up and the action started (the official MathsJam start time of 7pm is also the start of off-peak travel on the tube, so people tend to arrive later). We had about ten people in all, down from thirty-odd at January’s, when we took over the whole upstairs of the pub. There’s been a good mix of people in various walks of life, though most (but not all) had (or are doing) maths or computer science degrees: but everyone likes puzzles and games. This isn’t a full round-up: people sometimes split off into smaller groups, so it’s hard to keep track of everything, and there’s lots of chit-chat along the way that I haven’t documented.
♥ Someone autobiographically wondered what the chances of having two fire alarms in a day is.
♣ People were concerned when I brought out this noughts and crosses tiling puzzle (Think Tac Toe from this puzzle series), worrying at first it might be a physical copy of game itself:
The only solution anyone found wasn’t one of the four given on the back of the box:
Though it seems an unlikely to occur in a real game, it is a valid game position, so the solution is valid.
♣ Because noughts and crosses was universally unloved, we suggested replacements: Sim was explained, and 3D tic-tac-toe was played (on a 4×4×4 grid).
◊ Can you fit five rectangles together to form a square, where the rectangle side-lengths are each of the whole numbers 1 to 10? How many ways are there?
♦ Can you fit all twelve pentominoes, and an additional 2×2 square into: an 8×8 square; or into a 4×16 rectangle. We didn’t have time to try this one, or have any pentominoes handy.
◊ A coin puzzle (I heard a long time ago from a family friend):
Get five coins, three of one type, two of another (preferably circular, and of noticeably different sizes). The aim, starting from this alternating set-up:
is to get collect the coins of each type together, but all still touching, exactly as below:
The only available move, is to slide a pair of adjacent, but different, coins (using two fingers) to the other side of a row. The pair jumps these other coins: it starts touching them on one side, and ends touching on the other. You aren’t allowed to swap the order of the pair of coins, and no pushing or shunting any other coins. The only gaps you should create are those where a pair of coins has left.
You should be able to get to the goal in exactly five moves. It can be surprisingly frustrating, but once you get the solution, you should be able to do it quick enough to show people exactly how to do it without giving them the answer: this can make it more frustrating.
Around the table, there was a general coin levy, but some people still had to use coloured dice. I think they may have been these non-transitive dice.
◊ We had a game with a row of 10 coins of various values, and the two players took turns to pocket one of the coins from either end. After they’re all gone, the winner, as in much of life, was the person who gained the most money. The challenge was to find a general winning-strategy. An extension to three players was considered. You can find a solution and short analysis here.
♠ Some people were set this classic jumping puzzle using counters:
Here’s a flash version of this frog-jumping puzzle (is there another name for it?).
♥ Someone tried to remember a similar two-dimensional game that involved taking turns to move a piece of your colour forward into an empty square towards the opposite side, with the opportunity to block your opponent, but the rules we tried didn’t really seem to work well. The winner is the person who gets all their pieces to their far end first. (Let me know if you recognise this game).
♥ A couple of us briefly failed to remember the exact statement of Conway’s soldiers statement and the beautiful proof about how far they can jump.
♥ There was also some mention of a Martin Gardner ring or coin puzzle (the word “progresses” appears in my notes?).
◊ Calculator race
If you have a class of students, get them to pick two different three-digit numbers: 119 and 528 for example. Then, you pick another: say, 471.
Now the teacher must race against the students, who are armed with calculators, to find the sum of the products:
The teacher should be able to win every time, without being a mental arithmetic champion: how?
♥ The calculator race inspired a mention of the Wrongulator, a joke calculator that gets every operation slightly wrong (in a predictable and sometimes not very subtle way). A suggestion was that teachers should keep one handy to lend to students who forget theirs.
♠ I set the sum and product conversation puzzle, and though I explained the first few steps of reasoning (which I really like), some were disappointed that I didn’t remember the answer. I probably should have given the 20 version instead of the 100 version.
◊ Harry, Ron, and Gilly Weasley are taking three ogres across the river to Hagrids hut for their Care of Magical Creatures class. A non-magical boat needs to be rowed back and forth (ogres can row too). They all start on the same side of the river, but the ogres will gobble the students up if ogres ever outnumber the young wizards on either side. Additionally, if Harry and Gilly are ever left alone together, they’ll make out (even in front of the ogres), and Ron doesn’t want to see his little sister in that situation from across the river. So please don’t let that happen either.
Here’s a flash version involving nuns and ogres (as it is more traditionally told).
♥ The more traditional (fox, chicken, grain) river-crossing puzzle gets referenced in a joke in the very first episode of the BBC3 puppet comedy Mongrels. We got onto the topic of the Futurama theorem and there was a break-off conversation about computer memory, and using empty memory to swap variables. [Incidentally, it is possible to swap two variables without using a third dummy variable: this may well have been mentioned, but it’s a good puzzle even if it wasn’t.]
You can also read about what our alter-egos in Newcastle [edit: and in Manchester] were getting up to at the same time in their MathsJam (or in Leicester a month earlier). If you don’t have one nearby or can’t make it, check the #mathsjam Twitter hashtag (you don’t need to be on Twitter) for a feed of puzzles on and after the penultimate Tuesday of each month. Or start a Mathsjam nearby!