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Cutting my birthday cake

Guess what? Today’s my birthday. I’ve invited my friends, I’ve got the cake, and I’ve blown out the candles. There’s only one thing left to do: cut the birthday cake. As I pick up the knife, ready to cut the cake for my hungry guests and me, a question suddenly pops into my head: what is the maximum number of pieces I can get by cutting my cake $c$ times?

Well, first things first, there’s loads of ways you could cut your cake. So our first ingredient for the answer to this question will be a handful of assumptions. In this blog post, I’m going to chop up a square cake with a normal knife. All cuts will be perfectly straight, and will go across the entire cake (so we won’t be starting from its centre). Also, the pieces you get after cutting the cake do not have to be the same size! And finally, we’ll pretend our cake is a two-dimensional object—a humble square.

Let’s suppose that $p_c$ is the maximum number of pieces you can obtain by cutting your cake in $c$ slices, and see what happens for different choices of $c$.

Start with the easiest case where $c=0$. In this case, you haven’t cut your cake yet, so of course you still have the whole cake. This is a single piece in its own right, therefore zero cuts give you one piece only ($p_0=1$).

Next, make a long, straight cut across your cake (this is $c=1$). I guarantee you will cut your cake into two, ie $p_1=2$.

Let's make one big cut down the middle

Let’s make one big cut down the middle

Now we’ll cut the cake a second time ($c=2$). Your best bet is to make sure the second cut passes through both pieces, so you end up with four pieces. In other words, $p_2=4$.

Now let's make a second cut

Now let’s make a second cut

Time for a third cut. It turns out you can obtain a maximum of seven pieces ($p_3=7$).

And a third cut...

And a third cut…

Now, if we make one more cut (ie $c=4$), how many slices can we make? The answer is in fact eleven. Don’t believe me? Have a look at the picture below and count up the bits for yourself

A fourth cut gives us 11 pieces... count them!

A fourth cut gives us 11 pieces… count them!

Let’s summarise what we know so far. As shown in the graph below, our sequence (starting from $c=0$) is $p_c=1,2,4,7,11,$ etc. This sequence does indeed have a formula, but before I reveal it, see if you can spot the pattern. Give up?

The number of pieces follows a familiar sequence...

The number of pieces follows a familiar sequence…

The best way to see the pattern is to take one away from each term in our sequence, we are left with $0,1,3,6,10,$ and so on. That’s right: these are the triangle numbers! Remember, since we started our sequence from $c=0$, we also have the zeroth triangular number, which is nothing but zero.

Recall that the $c$th term for the triangle numbers is $c(c+1)/2$, so we can write out a formula for the number of pieces we get from $c$ cuts. It’s…\[\frac{c(c+1)}{2}+1.\]This expression will work for any $c=0,1,2,3,…$. If we try the formula for, say, $c=8$, then the formula tells you that you can get at most 37 pieces. So, the next time you have 36 friends at your birthday party, see if you can make one piece each for you and all your friends in just eight cuts!

But why does the formula work? Perhaps the easiest way to begin tackling this question is by writing down the differences between two consecutive terms. We get the numbers 1,2,3,4,…. More generally, the difference from $(c-1)$ cuts to $c$ will be just $c$. Anything familiar about these differences? They match up perfectly with the triangle numbers, and you can obtain $p_c$ by adding up all the first $c$ differences and one. In other words,\[p_c=1+\sum_{i=1}^ c i=1+ \frac{c(c+1)}{2},\]which we expected anyway. So we have proved the general formula. But why do we get these differences? To find out, we are going to gather some intuition, so let’s take a step away from the sequence and back to the cake. Suppose you just did $(c-1)$ cuts. If you want to maximise the number of pieces after your next cut, the trick is to line up your knife so that it will pass through all $(c-1)$ cuts exactly once. By doing so, you will cut your way through $c$ pieces, splitting each bit into two smaller slices. Hence you will end up with $c$ new pieces.

Our formula is for a square cake. Actually, it also works if your cake is a circle. But will the formula work for any two-dimensional cake you could think of? The answer is no; it turns out that the formula is only useful for convex shapes. As a rule of thumb, if your cake is not convex, you can look forward to even more pieces of cake! For example, it is possible to get as many as six slices from a crescent-shaped cake in two cuts. Try it yourself! If you haven’t got a crescent-shaped cake, sketch a crescent on a piece of paper and draw straight lines on it instead.

That’s one puzzle to try out. How about a few more…?

  • What happens if the cuts don’t have to be straight? Do you still get the same formula for $c$ cuts, or will it be different?
  • Earlier I assumed that the individual pieces you get after cutting do not have to be the same size. It is quite easy to make the pieces the same size for one or two cuts, but can you do it for, say, three cuts?

But that’s enough talk for today. Now, where’s that cake?

Happy birthday to me...

Happy birthday to me…

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Issue 05

Our latest edition, Issue 05, is available now. Enjoy the articles online or scroll down to view the magazine as a PDF.

Features

Fun

Printed versions

or

Download Issue 05 as a PDF

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In conversation with Bernard Silverman

It’s been said that a degree in mathematics opens many doors, but to many this might seem a slight exaggeration. Bernard Silverman, however, is an excellent example of a mathematics graduate who has indeed done it all. Silverman is currently the chief scientific advisor to the Home Office, a statistician, and an Anglican priest. These are just a few examples of his many achievements, starting from the gold medal he won at the 1970 International Mathematics Olympiad—the only person to do so from the western side of the iron curtain—at the beginning of his mathematical career. He went on to read mathematics at university, and eventually obtained a PhD in data analysis in 1977. “I was always interested in maths, but as time went on I became keen on doing it in a way that has applications in different things, and that is what drew me to statistics.” He jokingly adds that he felt he was never good enough to be a pure mathematician. In the course of our conversation with him, he took us on a journey through the diverse areas in which he has applied his statistical approach. Continue reading

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Linear algebra… with diagrams

A succinct—if somewhat reductive—description of linear algebra is that it is the study of vector spaces over a field, and the associated structure-preserving maps known as linear transformations. These concepts are by now so standard that they are practically fossilised, appearing unchanged in textbooks for the best part of a century. 

While modern mathematics has moved to more abstract pastures, the theorems of linear algebra are behind a surprising number of world-changing technologies: from quantum computing and quantum information, through control and systems theory, to big data and machine learning. All rely on various kinds of circuit diagrams, eg electrical circuits, quantum circuits or signal flow graphs. Circuits are geometric/topological entities, but have a vital connection to (linear) algebra, where the calculations are usually carried out.

In this article, we cut out the middle man and rediscover linear algebra itself as an algebra of circuit diagrams. The result is called graphical linear algebra and, instead of using traditional definitions, we will draw lots of pictures. Mathematicians often get nervous when given pictures, but relax: these ones are rigorous enough to replace formulas.

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Variations on Fermat: an agony in four fits

Fermat’s Last Theorem has been a source of fascination and the motivation for an enormous amount of mathematics over the last few centuries, both in attempts (eventually successful) to prove it and as the inspiration for other related questions.

This is the story of how an algebraic question inspired by Fermat’s Last Theorem morphed into an analytic question, which subsequently turned out to be expressed best as a geometric question, which could be answered using basic methods of plane geometry.

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Slide rules: the early calculators

Believe it or not, I never leave home without my trusty slide rule in my pocket. In the years before 1970, this would have been totally normal for any engineer, with the slide rule being the archetypal symbol of the engineering profession, much like how the stethoscope remains that of the medical profession. However, with the advent of the pocket calculator, the slide rule has completely vanished from public view and its demise is a wonderful example of a paradigm shift, as described by Thomas S Kuhn in The Structure of Scientific Revolutions.

I often get asked “What are you doing with that thing?” when I grab my slide rule to convert miles to kilometres or, once I’ve finished refuelling, to calculate my fuel economy. Most of my students have never seen a slide rule and are at first quite incredulous when I show them my Faber–Castell 62/82N. Not that I’ve ever managed to convince one of them to switch to a slide rule, but at least I normally manage to instil some interest in them for these mathematical instruments.

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Seven things you didn’t notice in Issue 04

With just a few days to go until we launch issue 05, we thought it’d be fun to share a few bits and pieces that we hid around issue 04. If this gets you excited for issue 05, why not come to the launch party on Tuesday?!

Scorpions

Since we published the horoscope in issue 03, scorpions have been running around all over Chalkdust HQ. Three of them managed to sneak into issue 04.
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Chalkdust Issue 05 coming 7 March

We are pleased to announce that Chalkdust Issue 05 will be released on Tuesday 7 March. To celebrate this, we will be holding a free launch event on Tuesday 7 March in the Print Room Café at UCL from 7.15pm. We’d love to meet more of our readers, so please come along and invite your friends. If you can’t make the event but you’d still like to get your hands on some copies of Chalkdust, please see below for more information.

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To share, or not to share

The Montagues and the Capulets have never been friends and Juliet is quite aware of this. Even at her young age, she knows that her love for Romeo is an impossible dream her father will never accept. So, she designs a strategic plan. She will take a couple of sleeping pills, just enough to make her look like she is dead to trick everyone into thinking that she has passed away. Brilliant! If everything goes right, she will always be happy with Romeo… but if things go wrong… well, you never know. Continue reading