The first MathsJam of the year was well attended. Despite not being on our usual table (there was no jazz band on this week, so we were allowed a bigger table further into the pub) everyone found us ok, and a few people brought baked goods – always a precursor to an excellent MathsJam.
We started off with some quick mental arithmetic brainteasers: how many straight cuts do you need to make to slice a flat square cake into 196 equally sized square pieces? Several people got the answer quite quickly, while others tried to cheat by stacking cake pieces and moving them around between cuts. No cheating!
Various people’s mathematical Christmas presents were present, including a tangram puzzle, a set of ‘Crazy Four’ in which you have to arrange four cubes in a line so that each face has the same colour all the way along (frustrating!), and a ridiculous Guinness World Record Domino Challenge set which was given to some of the Domino Computer volunteers as a hilarious sarcastic Christmas present, given that they’d already stacked enough dominoes to never want to see one again. The set claimed that the world record for placing horizontally flat dominoes in a vertical stack, one on top of the other, using one hand, was 19 dominoes in 30 seconds. We managed to beat this in a pub on a slightly wobbly table, so we’re not sure if this is a world record that will stand for long. It doesn’t appear to be listed on the Guinness website, although the Amazon listing for the product confirms in the description that the record is 19, as does the box. We concluded that this was just a ploy to sell Christmas gifts, especially as the record was apparently set in the Guinness World Records office by someone who presumably works there. It was also a good excuse to explain the Domino Computer to anyone who hadn’t heard about it, and relive former glories.
We discussed several problems from Futility Closet, an increasingly useful reference for MathsJam puzzles, including one about soldiers with different injuries (we agreed it needed restating in a less depressing way, possibly involving cake) and one about a belt around three pulleys, which took some people a while to realise there’s a nice shortcut.
We also tried to work out the trick in this probability teaser, in which the following game is played:
You’re about to play a game. A single person enters a room and two dice are rolled. If the result is double sixes, he is shot. Otherwise he leaves the room and nine new players enter. Again the dice are rolled, and if the result is double sixes, all nine are shot. If not, they leave and 90 new players enter.
And so on, the number of players increasing tenfold with each round. The game continues until double sixes are rolled and a group is executed, which is certain to happen eventually. The room is infinitely large, and there’s an infinite supply of players.
The puzzle asks, if you’re selected to enter the room, how worried should you be? Not particularly: Your chance of dying is only 1 in 36. Later your mother learns that you entered the room. How worried should she be? Extremely: About 90 percent of the people who played this game were shot.
It took us some time to work out what the difference was in the information your mother has, and the information you have, since each of the conclusions is valid from the perspective of the relevant person. It’s a nice illustration of how counterintuitive probability can be!
There was a good discussion of this train puzzle, which some people had seen before and others worked out slowly – which descended eventually into a discussion of countable and uncountable infinities, and why infinity squared (if you’re allowing it to be a thing) is the same as infinity. Minds were blown, cake pops were eaten.
We experimented with making Menger Sponges from cards (a Manchester MathsJam staple) although this time Katie had printed some cards up with a Sierpinski Carpet design, which meant the sponge pattern went down to extreme levels of detail. Since we were limited in time and ability to be bothered, we made single cubes, although the cards could easily be used to cover the outside of a sponge made of 20 cubes.
Ross, who’s been a recent addition to Manchester’s attendance, brought a puzzle he’d invented, involving six dice. You roll the dice, and then re-roll any subset of the dice which contains duplicates (so, for example, rolling 134466, you’d re-roll the 4s and 6s), and then repeat this process, looking at the whole set each time for duplicates. If you ever find you have 6 unique values (1 to 6), that means you win, and if you ever get to a point where you’d have to re-roll all 6 dice (e.g. if they’re in three pairs, or two triples, or all the same) then you lose. Are you more likely to win or lose?
Various on-paper calculations were done, several confident-sounding fractions were thrown around, and the puzzle was discussed at other MathsJams including the newly-restarted Washington DC MathsJam, which took place a few hours after we’d finished and gone to bed. I’m still not sure what the answer is, but I’ll work it out when I get a minute.
On a night when several other MathsJams were cancelled or curtailed due to the horrible weather, I’m glad we managed to get together and it was good to see both old and new faces! And cake.