This month saw a record high turnout, requiring as many as three tables being pushed together, a whole bag of maltesers and a tin of shortbread someone got for Christmas and hadn’t eaten yet. We also had one new attendee who had previously been a regular at Newcastle MathsJam, and has now moved to Manchester for a PhD. Not that it’s a competition or anything, but in your face Newcastle. In fact, the turnout was so large that I couldn’t even keep track of everything that was going on, and when I collected in all the scrap paper I found people had written down several things I wasn’t aware we talked about, including the method for cube rooting large numbers used by Maths Busking.

We started off with some puzzles I found on a Canadian maths challenge for school kids – the nicest of which was, what’s the smallest multiple of 225 which can be written in decimal using only 1s and 0s? This puzzle took some of us a couple of minutes and others longer, but there’s a nice intuitive step which leads you to an answer.

We also discussed a problem which came through from Twitter, where there are 100 seats on a plane and everyone is assigned a seat. The first person to board the plane has lost his ticket, and sits in a random seat. Each subsequent passenger sits in his own seat if it’s available, and otherwise sits in a random seat. What is the probability that the last person to get on sits in his own seat?

Someone on Twitter pointed out that the sum of the first three triangular numbers is the fourth triangular number, and the sum of the next four is the sum of the two after that, and so on. I’m halfway to a proof of this pattern continuing, so please don’t spoil it for me, but I have a general formula for the statement and just need to plug in some horrible quadratics. EDIT: I have given up on the horrible quadratics as they became horrible quartics and I wanted to eat my tea.

Edinburgh MathsJam stated that as in Edinburgh exact change is needed for buses, one of their attendees had found themselves in the situation where they had £1.30 in change but couldn’t pay their fare of £1 exactly. What combination of coins were they holding? And once you’ve worked that out, what’s the most money you could be holding and still not be able to pay £1 or any multiple of £1 exactly?

Our new member from Newcastle had brought along one of Professor Ian Stewart’s books of random (in the non-mathematical sense) mathematical curiosities, so we had a look at this one: Can you find two numbers for which the sum of the squares of the two numbers is the result of writing the two numbers side by side? Paul, after some frantic and inspired calculator bashing, found that $27^2 + 45^2 = 2754$, which is frustratingly close to working, but otherwise no-one got an answer. Can you?

The other thing discussed was domino logic gates – as regular readers of the site may have learned, Matt Parker is planning an ambitious domino-based computer for Manchester Science Festival, and the previous weekend had seen several Manchester MathsJam regulars join a group of volunteers to test out the logic circuit design and the practicalities of large scale domino runs. Needless to say, the process is somewhat nerve-wracking, especially when the computer is nearly finished, as any minor knock could set the whole thing off early and require a major reset.

The team‘s nerves had just about calmed down by Tuesday and we chatted about some of the issues we’d faced. To build a working logic circuit, we’d often need a way of setting up a junction so that if it gets set off from one of the inputs, the toppling continues along another input but crucially doesn’t set off the third (back along the other fork). Our main domino computer trial had been marred by knock backs, setting chains off the wrong way and some chains failing to fire, so designing a reliable junction was important. One third of the team had taken a pile of dominoes with them and had come up with a solid design, which (mildly controversially) involved a domino standing on its long edge, but had stood up to repeated testing without misfires or knock back. I’ve attempted to approximate it in this photo, but this doesn’t do justice to the precision and engineering, and precision engineering, that have gone into the design.

As the theme here is puzzles, can you think of how dominoes could be used to create an OR gate (easy), an AND gate (not as easy) or an XOR gate (can be done quite elegantly!)? Matt Parker’s design for an AND can be found here. To find out more about Matt Parker’s ambitious domino computer, check out the links in this sentence or wait for the inevitable write-up on this site. For now, here‘s a very exciting and revealing sneak preview video, in which we test a length of domino run for timing purposes so we can use it to build in delay runs of the correct length. The video features a guest appearance from Matt Parker’s shoe, and pointing hand.

A lot of fun puzzles that night, and that domino computer is mindblowing. Even at the early stage. It’s really impressive. Getting everything to sync up right must be a pain.

Btw, as the one who brought the shortbread, I will point out that it was a fresh tin I bought for mathsjam. Shortbread would never last 9 days around me, let alone the 9 months from Christmas! :P (Though, I’m sorry I forgot to pick the tin up, and my copy of set, when I left!)

$10^2 + 0^2 = 100$

One thing about the Ian Stewart problem; they have to be two digit numbers (otherwise 10 and 0 work, which is rather boring). There are two solutions, I think. If I remember rightly, you can represent the problem as $a^2 + b^2 = 100a + b$ and solve as a quadratic to get halfway there.

Anyway, fun was had. :)

12^2 + 33^2 = 1233

88^2 + 33^2 = 8833

In base-10, no b satisfying a^2 + b^2 = Concatenate[ab] has 9 digits.