One day, a couple of months ago, I was walking my son to nursery and he asked what I was doing that day. I said I was going to do some teaching. What about? he asked. Well.

# You're reading: Columns

### Ning Nang Nong Latin square

My son is obsessed with the Spike Milligan nonsense poem ‘On the Ning Nang Nong’. Here’s a video of Spike reciting it.

This weekend, he asked me to help him learn it. I’ve tried to memorise it before, to save having to find the book when he wants me to recite it. But somehow, it’s never quite stuck. I can remember all the bits and the basic order (Cows-Trees-Mice), and know what happens after the lines ending “Nong” (“Cows go bong”), “Ning” (“Trees go ping”) and “Nang” (“Mice go clang”). What I struggle with is remembering which order the “Ning”, “Nang” and “Nong” go before the one that rhymes with what comes next.

At the weekend, I wrote “Ning”, “Nang” and “Nong” on pieces of paper and we rearranged them as we read the poem. I realised my difficulty is a mathematician pattern-spotting one. There’s a not-quite Latin square embedded in the poem.

### Making Tricurves

*Tim Lexen has written a series of posts on the topic of Tricurves: Bending the Law of Sines, Combining Tricurves and Phantom Tiling. In this latest post, Tim has been working with our own Katie Steckles to turn Tricurves into real objects to play with.*

When you discover an interesting mathematical shape or object, there’s a strong instinct to play with it – maybe by drawing sketches and doodles to test the limits of the idea. But in the case of Tricurves, drawing an accurate shape takes a little time, and it doesn’t lend itself well to idle experimentation.

Producing a physical version of a shape, in enough quantity to allow for experimentation, makes it much more tangible. In our own respective locations, we’ve each made use of laser cutting facilities to produce wooden Tricurve tiles to play with, and we encourage you to join in.

### Nine months of HLF blogging

I’ve now been writing fortnightly blog posts for the Heidelberg Laureate Forum’s Spektrum blog for over nine months. Small trumpet noises! Here’s what I’ve been writing about:

- Public Key Cryptography (29 January) in which I describe the famous work of HLF regulars Diffie and Hellman
- A puzzle ‘four’ the new year (11 January) in which I discuss number puzzles, which do and don’t involve the digits 2, 0, 1 and 9
- The Twelve Facts of Christmas: Pascal’s Triangle (20 December), a festive round-up of mathematical facts
- Right on the Money (4 December) in which I wonder if the new British £50 will have a mathematician on it
- Measuring the Change (21 November) on the forthcoming redefinition of the kilogram
- Permutations and Tribulations (13 November) on the superpermutations discovery made anonymously on the internet
- Katherine Johnson (7 September) on life of the mathematician
- Game, SET and maths (16 August) about the card game SET
- New shape discovered – right under your nose (6 August) in which I describe the newly-discovered scutoid shape, found in nose cells
- Flexagons and false advertising (27 July) in which I explain how ordinary paper can be much more impressive if you flex it, Martin Gardner-style
- Few tile attempts (5 July) in which I discuss tilings and tessellations, and then decide how to redecorate my bathroom
- Cake and Coincidence (21 June) in which a striking coincidence prompts a discussion on probability and birthdays
- The Bridges of Königsberg (29 May) discussing the famous problem in graph theory
- A truly special occasion (18 May) all about myself and my partner’s Golden (ratio) wedding anniversary
- Colouring in like a mathematician (3 May) covering the then-current Chromatic Number discovery

I’ll continue writing posts for HLF’s Spektrum blog as long as they want me to – keep checking their Twitter feed or the blog’s RSS feed to see them as they appear!

### Carnival of Mathematics 166

The next issue of the Carnival of Mathematics, rounding up blog posts from the month of January, is now online at Math With Bad Drawings.

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

### Pringle stack mathematics

Pringles ran a Super Bowl advert. In case you’re looking for ways to give Pringles more money, apparently you can buy several tubes of Pringles and mix the flavours. (Pringles are a type of food. Super Bowl is a kind of sport. None of that matters, what matters is…) The advert shows a man stacking three Pringles together and claims there are 318,000 possibilities.

### I’ve made myself a calculator

For the past couple of weeks at work, I’ve been checking through our computer-based assessments before the students have a go at them. That means I’ve had to do lots and lots of calculations by hand, to confirm the computer’s got the right answer. Well, not quite by hand – I use a calculator for the stuff that I can’t keep in my head.

I’ve got a calculator app called RealCalc Plus on my phone, which I highly recommend. The main thing it’s got going for it is its RPN mode. While most calculators ask you to type in expressions pretty much as you’d see them on a page, read from left to right with numbers separated by operators and brackets, reverse Polish notation looks like this:

1 5 √ + 2 ÷

That computes $\phi = \frac{1+\sqrt{5}}{2}$. You put the numbers in first, and then say what to do with them. This might look obtuse, but it can make long expressions much easier to type in. Because there are no brackets, you don’t need to remember to close them. And if the same fragment appears more than once in the expression, you can just duplicate it on the stack, instead of typing it all out again. I’ve been using an RPN calculator for a few years now, and I get a warm mathmo feeling when I think about all the time it’s saved me.

But there are a couple of things that regularly trip me up. Once you’ve performed a calculation, you can see the result but you can’t see how you got there, so when you’ve got a couple of long numbers sitting on the stack and you can’t remember which is which, you just have to start again. And if you want to repeat a calculation but with a slightly different starting value, you’ve got no choice other than to type it all in again.

So, about halfway through my marathon of testing, with thumbs sore from tapping calculator buttons, I decided there was nothing for it except to make my own calculator. It would mainly be like RealCalc, but do something to solve those two problems. I headed straight to my favourite gaudily-decorated coding environment, Glitch.com, and set to work.

In this case, I think the day or so I spent making the calculator was worthwhile. I’ve been using the new calculator to finish off my testing duties, and it feels much better to use: I can do calculations quicker, and I lose track of what I’m doing less often.

The first change I made was to display not just the results of calculations, but how they were obtained. When you press `1 2 +`

, a box is pushed to the stack with a 3 at the bottom, but also the 1 and 2 and a + symbol above it. This works for nested operations, too, but the ingredients are hidden by default until you tap the box. There’s an ‘undo’ button, which throws away an operation and puts the operands back on the stack – very handy when you tap the wrong operator!

Numbers that you type in directly are shown in blue. You can tap any blue number and enter a new value, and any operations it feeds into are recalculated. I’ve used that quite a few times to set up a formula, such as the quadratic equation, and each time I used it I just need to change the input values and the result appears at the bottom.

It’s also useful for factorising numbers: I can start by typing something like `10199 3 ÷`

, and I just replace the 3 with different prime numbers until the result is an integer.

After a while, I realised that there was another optimisation to be made: sometimes a formula uses the same variable more than once! For example, in the quadratic formula, $b$ appears twice:

\[ x = \frac{ -b \pm \sqrt{b^2 – 4ac}}{2a} \]

It would be nice if you could use the same number in more than one place, and have every instance update when you change the value of one of them. Every RPN calculator has a ‘copy’ button, to push another copy of the last item on the stack, so I just decided to make copies remember they’re linked. You can move copies of a number around, but when you overwrite one of them, all the others take the new value too.

In order for this to not make your head explode, I needed a way of showing which number boxes are linked together in this way. Asking the user for a name felt like overkill, and would interrupt your flow unnecessarily if you’re not planning on using the overwriting feature.

I came up with a nice solution: when you copy a number, it’s assigned an emoji. All copies of the number have the emoji stuck onto them, so you can quickly tell who they are.

This was way more useful than I expected! While I was testing a horrible question involving cancelling fractions with very large denominators, the linking feature really came into its own. By setting up $n/p$ and $d/p$, I could quickly find common factors of $n$ and $d$ just by changing the value of $p$ until both divisions produced integers. Bonza!

I’m quite happy with my calculator and I haven’t felt the need to go back to RealCalc yet. Hooray!

You can use it too: go to nice-calculator.glitch.me. If you open it on your phone, you can add it to your home screen and it should act like a normal app, instead of a web page.