January’s MathsJam was a bit massive. It’s now a week later and I’ve only just gathered enough thoughts together to do this writeup.

There were nine of us this month, all but one of whom either maths students or lecturers. A major theme of the night was of professional mathematicians or nearly-professional mathematicians forgetting basic high-school methods. This led to quite an intense session of puzzling and proving.

Things didn’t start out that way, though. A few weeks ago I found the website of a mathematician in Illinois called Alan Schoen, and his page about Lominoes. They’re a pretty interesting set of shapes! I ordered a couple of sets and they arrived just in time for the MathsJam.

IMAG0128 - christianp - Flickr

We spent a lot of time trying to build a ziggurat:

Lominoes ziggurat - christianp - Flickr

Because I had two sets, we split into two teams and the competition got a bit serious. The spirit of friendly cooperation was away on a business trip.

Team A (best team, my team) did a lot of writing and calculations and thinking. We hardly touched a piece other than to measure it.

IMAG0129 - christianp - Flickr

Team B, on the other hand, were much more hands-on in their deliberations. They got quite emotionally involved with the pieces – when I asked why a particular pile of pieces wasn’t being touched, I was informed that those were the “bad” pieces.

A key point of contention was the question of which pieces needed to be used. I think we settled on everything except the greys and half the blues.

We spent well over an hour trying to build a ziggurat. Neither team succeeded: both teams failed. While packing the pieces away, Andrew (Team B, free spirit) constructed this challenging artwork:

IMAG0130 - christianp - Flickr

Matthew Taylor again posted a puzzle on Twitter before the MathsJam night. This time it was a question about points with integer coordinates on the boundary simplex-ish shapes. We didn’t find a solution.

A Fibonacci fact

David Cushing showed me this fact. It’s pretty cool!

Draw an $N \times N$ grid and fill it with the Fibonacci numbers:

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Pick any $2 \times 2$ square, and work out its determinant. For example,

picnikfile_aXjTcv - christianp - Flickr

\[89 \times 8 - 55 \times 13 = -3. \]

What’s the pattern?

The others at MathsJam spotted what was going on with a little bit of prompting. Click here if you’d like a spoiler.

The Dorset MathsJam posted a problem to twitter:

This is pretty similar to the lights-out puzzle we discussed in December. It turns out that Stephen, one of the new attendees this month, had a much better proof that not all patterns in the lights-out puzzle could be produced: all of the moves change either zero or two corner squares, so patterns with an odd number of corners lit up can’t be produced.

A bug on a boinger

Andrew posed this conundrum:

I have a $1$ metre elastic band tied to my waist. I tie the other end to a wall and start walking away at $1$ metre per second. There’s a bug on the wall which decides to walk along the elastic band, at $10$ centimetres per second. Does it ever reach me?

After you’ve had a think, click here for the answer.

Emboldened by his solution to the last puzzle, Ji posed this question

Silly people chasing each other

There are $5$ people spaced evenly around a circle. Each person is chasing the person to their right, taking the shortest route instead of walking around the edge of the circle. They’re all moving at the same speed. What happens?

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Working out what happens shouldn’t take too long. Working out when it happens involves a bit of geometry. If you’d like to see the answer, click here.

Andrew had another puzzle. It had more of us stumped than the England cricket team in their recent test matches.

The worst sequence in the world

\[ \dots, 7, 9, 12, ??, 24, 36, 56, 90, \dots \]

Andrew quite rightly tried to avoid giving us many hints about this. When I got it, I felt like slapping him in the face. You need to experience this feeling too.

Click here for the first hint.

There was a brief break from puzzling when I told everyone that I was taking notes for a Mathematics Today article, and I wanted to hear how everyone was feeling.

Several of the undergrads said that they’d spent four years studying maths and thinking they were pretty good, until they came to MathsJam and had a go at some of the puzzles. They were happy about that fact. I think Andrew and I do give them some very hard puzzles precisely because they’re keen undergrads, though.

We were all agreed that MathsJam keeps you on your toes – “Why don’t I know this? It’s first year stuff!” was a common exclamation throughout the evening.

Matthew asked a couple of questions about PhD interviews – in particular, how formal they are. Answer: not very. They’re mainly for you and your supervisor to see how well you’ll get along.

This led to a discussion of tropical maths because Matthew had visited a potential supervisor about a project in that field.

Tropical maths

In tropical maths, the addition and multiplication operations are different to the usual ones. They’re called $\oplus$ and $\otimes$ and they work like so:

\[ \begin{align} x \oplus y &= \min(x,y) \\ x \otimes y &= x + y \end{align}\]

So $3 \oplus 4 = \min(3,4) = 3$ and $3 \otimes 4 = 3 + 4 = 7$.

Apparently you can solve sets of simultaneous equations with these operations. I just made one up, and it works:

\[ \begin{align} x \oplus 2y &= 1 \\ 3x \oplus y &= 2. \end{align} \]

In the usual arithmetic, that means:

\[ \begin{align} \min(x,y+2) &= 1 \\ \min(x+3,y) &= 2. \end{align}\]

But I’ll stick with the tropical notation. My first thought is to multiply the first equation by $3$ so that the coefficients of $x$ are the same in both equations:

\[ \begin{align} (3 \otimes x) \oplus (3 \otimes 2 \otimes y) &= 3 \otimes 1 \Leftrightarrow \\ 3x \oplus ((3+2)y) &= 3+1 \Leftrightarrow \\3x \oplus 5y &= 4 \end{align} \]

So the simultaneous equations now look like this:

\[ \begin{align} 3x \oplus 5y &= 4 \\ 3x \oplus y &= 2 \end{align} \]

or,

\[ \begin{align} \min(x+3,y+5) &= 4 \\ \min(x+3,y) &= 2 \end{align} \]

It follows that $x+3 = 4$, so $x=1$, and $y = 2$. How interesting! Abstract algebra is the best.

There was some talk about Golomb rulers, which have the property that any length can only be measured at most one way. A similar question is the smallest set of weights you need to be able to measure any whole number of units on a set of scales – it’s powers of $2$ if you can only put weights on one side, and powers of $3$ if you can put weights on either side. We wondered what setup would use powers of $4$.

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Ji kept trying to get me to think about how to fit $72$ T-shaped tetrominoes (tetrominoids?) together into a $6 \times 6 \times 6$ cube. I had no idea where to begin. Hopefully in February he’ll bring some along and explain what he was on about.

Andrew mentioned a question that the London MathsJam had apparently discussed in November and December but not found a satisfactory answer to.

Maximising your profit from an awkward benefactor

Somebody presents you with two envelopes containing money. One envelope contains twice as much money as the other. You’re asked to pick one envelope. It turns out that the envelope you pick contains £10. You’re offered the chance to switch to the other envelope. Should you?

Andrew and I were fairly satisfied with our answer, but the others either weren’t convinced or had come to different conclusions. I’ll let you discuss this in the comments thread. It’s a rather finicky question.

As we were just about to get ready to leave, I mentioned that while writing an edition of the Internet Maths Aperiodical I had found out that there are $85$ ways to tie a tie, $43,200$ ways to do up your shoelaces and (at least) 25 ways to wear a scarf.

So the question naturally arose:

How many ways are there for a banker to dress himself?

We came up with a fairly extensive list of configurably-worn items:

  • Tie
  • Shoelaces $\times 2$
  • Scarf
  • Shirt – buttons done/undone, map buttons $\rightarrow$ buttonholes, inside-in or inside-out, sleeves rolled up or not, etc.
  • Placement of phone, wallet, keys in pockets
  • Order of notes and cards in wallet

We thought this provided enough combinations to allow every banker who has ever lived to dress uniquely. So why are they not‽

Finally, someone asked if David and I had made any more progress on the dragon-catapult puzzle mentioned last month. I regretted to inform them that we’ve decided it’s unsolvable in too many cases to be interesting. Someone, possibly Tash, asked what would happen if, instead of firing a catapult at the dragon, we had to chase it on foot. This is an old problem, most often stated in terms of lions trying to catch gladiators.

Very finally, Matthew asked if you can prove anything non-trivial in a tweet. The next day, he tweeted this:

And that was the end of January’s MathsJam! It was a marathon. I was intellectually exhausted for days afterwards. I’m going to try to take some games to play next month, so we don’t do quite so many difficult puzzles. We need more amateur mathematicians, too!