Let them share cake

There comes a point in every person’s life where they have to learn how to share fairly. Admittedly some people seem to ignore this point, sailing on through life gleefully seizing more than would be justified, but we’re willing to bet that the situation of having to divide up a resource (for example some food or a list of chores) into parts that everybody is happy with is pretty much universal.

If there are just two people who want to split the resource, then there is a simple method to ensure that it is divided fairly. This concept (called the “I cut, you choose” method) is so old that it’s even mentioned in the Bible. As the name suggests, the method involves one person splitting the resource into what they consider to be equal halves, and then the other person picking which (if any) of the pieces they think is worth more. The person who chooses is bound to be happy, and the person who cut can’t complain since they were supposed to divide the resource into equal pieces. The solution for two people, then, is so simple that it doesn’t seem like mathematics at all. However, the problem becomes significantly harder once you start to include more people — so difficult, in fact, that a completely ‘satisfactory’ answer for an arbitrary number of sharers was not found until 2016… Continue reading


Magic behind the Fibonacci sequence

In mathematics, there are countless sequences such as arithmetic sequences, geometric sequences, and many more. The Fibonacci sequence is one of them, but it is different from other sequences in that it can be easily found in everyday life. Let’s take a look at patterns that can be discovered in Fibonacci numbers and how we can find them around us.

In a Fibonacci sequence, every number after the first two numbers is the sum of the two preceding ones.

0, 1, 1, 2, 3, 5, 8, 13,…

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Ten things you didn’t notice in Issue 06

Next Tuesday, Chalkdust issue 07 will be released (don’t forget to book your free ticket to our launch event). To help you to get as excited about the launch as we are, here are some of the things we hid in issue 06.

1. A Mary Poppins reference

Hopefully this one got you singing.

2. Sophie Willock

Sophie Willock wrote a letter to Dirichlet. Who is Sophie Willock, you may be asking? Turns out it’s Sophie Bryant before she married.

3. Scorpions

It just wouldn’t be an issue of Chalkdust without scorpions everywhere.

4. Balanced ternary

Rob Low’s article on balanced ternary showed you how to write numbers in base 3, using 0, 1 and -1. During this article, the page numbers were written in balanced ternary

5. Grumpy Gauss

In our interview, Cédric Villani told us how he’d love to travel back in time to meet Gauss, although maybe Gauss would be too grumpy to talk to him. It was accompanied by this picture of him looking grumpy.

In the original portrait, Gauss is a lot less grumpy. Adam had a lot of fun playing with this portrait in Photoshop.

6. Crossnumber header

Once again, the crossnumber header was made with part of the previous crossnumber.

7. Crossnumber grid

At first glance, issue 06’s crossnumber grid looked like it had no symmetry. But after a longer look, you may notice that turning it upside down inverts the colours. Humbug had a lot of fun designing this grid.

8. A typo

A strong contender for the stupidest joke in issue 06 award.

9. LaTeX

Just to prove that we made the whole thing in LaTeX, we snuck a couple of \LaTeX{}s in.

Edit: It’s just been pointed out to me that I missed a thing. Here it is:

10. The number on the cover

This number in balanced ternary appeared on the cover. If you translate it into decimal, you get 5318008.

Hopefully these have got you excited about issue 07, where we have hidden even more things! Make sure you come along to our launch quiz to pick up your copy and start looking…


Sylvester’s convex hull problem in R

James Joseph Sylvester. Image: public domain.

James Joseph Sylvester was born just over 200 years ago on 3 September 1814 in London and was educated at Cambridge. However, as a Jew, he was awarded his degree only in 1872 when Oxford and Cambridge Universities abolished the theological tests for graduates. In 1838 he became professor of Natural Philosophy at University College London (the first English university to admit students regardless of race, class or religion, and also the first to admit men and women on an equal basis) where he had been, briefly, an undergraduate ten years earlier, and three years later was appointed to the chair of mathematics at the University of Virginia in Charlottesville, a post he held for a few months. In 1854 he was appointed to the chair of mathematics at the Royal Military Academy at Woolwich until his retirement, under military rules, at age 55. This was followed by a fruitful period as a mathematician. In 1877, Sylvester went back to the United States as professor at Johns Hopkins University, and, aged 68, in 1883 was appointed Savilian professor of geometry at Oxford. He retired in 1892 and died in London on 15th March 1897. The Royal Society instituted the Sylvester Medal in 1901 for the encouragement of mathematical research.

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Power-up review

Last year, Power-up: unlocking the hidden mathematics in video games by Matthew Lane (Amazon: UK, US) was released. I’ve done some writing and talking about maths and video games myself, so I was keen to read it.

What a great place to sit and read a book! Image: David Iliff, CC BY-SA 3.0

I took a copy of it with me on holiday to the Lake District, where I had plenty of time to sit in a garden and enjoy it.

And enjoy it I did! Power-up is a really well written and enjoyable book that takes you on a tour through many different areas of maths and their relation to many different games. I was surprised by just how many different games the book finds maths hidden in.

My favourite chapter of the book was The thrill of the chase. This chapter looked at the green and red shells in the Mario Kart games. As hopefully everyone knows, the shells in Mario Kart are weapons used to attach to the other racers: green shells travel in straight lines and bounce off objects; red shells travel straight towards an opponent, but they break if they hit other objects. You may be thinking to yourself: “What is the best angle to fire a green at?” or “What path will a red shell follow?” or “When will the shells hit the opponent?”. If you are, then you should read Power-up. If not, you should probably read Power-up anyway…

It’s worth noting how nice the book looks inside. Sometimes the maths in maths books can be badly typeset, which is at best distracting and at worst simply wrong. This book has no such problems; the equations are beautifully typeset:

Nice typesetting!

The only problem I can find with Power-up is that at times it talks about games that I’m less familiar with. These were less easy to follow, and less interesting to me, as I was unfamiliar with the problem the book was discussing. The majority of the games discussed, however, are very well known, so this problem is small. And perhaps the best solution to it would be for me to play more video games.

In case you’re wondering, other highlights of the holiday included meeting this famous cow, and climbing Place Fell. I highly recommend both the holiday and the book, although it may be possible to enjoy them both separately….


Modelling in a heartbeat

Today at Chalkdust, we’re celebrating our love for mathematics, and what better way to do that than by analysing our heartbeats? The heart is composed of four main chambers: two atria at the top and two ventricles at the bottom. Apart from carrying our hopes and dreams, the heart’s main function is to pump blood around our body. This can be split into two different circuits, the first one is low-pressure (in blue below) and pumps blood from our body to our lungs to get oxygenated before coming back to the heart. The second is a high-pressure circuit (in red) which then pumps this oxygenated blood out of the aorta and to the rest of our body. Continue reading


Math Blaster

Math Blaster is a classic game many of you will have played in your childhood. But did you know it was first released in 1983 for the Apple II and the Atari 8-Bit? The original version was created by Davidson & Associates as a simple zapping game with the aim of teaching school-level maths, though it didn’t have a storyline at the time. Since then, it has been re-released countless times for various consoles and computer.

And which version did you play? My one came out in 1998 and was called Math Blaster: Ages 6-9. But not too long ago, I discovered another version released in 1994 called Math Blaster Episode 1, which was designed for the Sega Mega Drive (that’s Sega Genesis for the Americans reading this blog post!) After hearing about it, I decided to relive my childhood, this time in 16-bit graphics. I never played Math Blaster on a console before, so I went for the easiest settings and as many lives as the computer would give me(!)

Setting the scene

The game starts with Blasternaut asking Spot to pop outside their spacecraft and repair one of the parts in the engine. Blaster looked as if he was a father sternly telling his son to tidy up his bedroom!

Spot reluctantly agrees, and starts his repair job, when all of a sudden, some yellow trash alien snatches Spot behind Blaster’s back! As soon as Blaster finds out, he calls the Galactic Commander and hatches a plan to save Spot. Apparently the first step is to…clean up the trash?

Blasternaut to the rescue!

Part 1: Trash Zapper

I saw some trash flying my direction; each bit of junk had a different number, and only one of those numbers fits the equation on the panel, so I needed to zap the correct one. It took a while for me to get the hang of the controls, probably because I never played on a SEGA console before! But even when I figured out how to aim and shoot, it was still difficult to zap anything because you had to move the cursor using the arrow controls. All I needed in the 1998 version was a mouse – so much simpler!

Now, where’s that seven?

You also have bonus stages where you shoot asteroids to gain points. It doesn’t affect your chances of rescuing Spot, but I like the graphics for the asteroids – it’s surprisingly realistic for a game running on only 16 bits!

Once the stage is over, you see the trash alien flying his spaceship into a crater on some planet, with Blaster in hot pursuit.

Part 2: Cave Runner

Blasternaut starts at the bottom of a series of caves with a 0 attached to his chest. As I played the 90s version, I recognised the drops of water (or whatever liquid it is) on the roofs of each ”floor” of the cave. Each drop will say something like “+2” which adds two to whatever number Blaster is carrying, or “-11” which takes eleven away instead, etc.

Also, there’s a gap you fly through to reach the next floor. Playing the 1998 version didn’t prepare me for how you get past it – I saw “3+0” and “17” and tried to get 17 so I could hit a lever and open the way through (like the 1998 version). When I did that I was zapped straight away! It turned out I needed a number between 3+0 and 17 to pass. So I started to chug my way through tons of drops.

The irritating part was when I wanted to take 2 away from the Blaster’s number and the “-2” drop didn’t appear, so I kept flying into lots of drops until I ended up with a number I was looking for. I passed through at last…only to fly into some bat-like creature!

There were many other times I lost control while I was getting Blaster to fly – when I wanted to pass one floor, he sometimes went through several (which was a good thing, actually!) There’s a few monsters flying/walking by here and there, but I thought it was quite easy to shoot them down.

It’s a bit cramped over here in the cave runner!

After I completed that stage, I saw the alien fly out of a second crater. Blasternaut hasn’t given up on chasing him…

Part 3: Classic Math Blaster

At this point, the trash alien has carried Spot into a UFO which is hovering mid-air. The UFO has four “holes” which are like entrances. By now, you’ll have figured that to save Spot, Blaster needs to sneak in through one of the holes. Sounds easy enough, right? Wrong!

Watch out for the low-flying litter!

Basically there’s an equation shown on the UFO. You have to dodge the trash (as well as any loitering monsters) and fly into the port that has the correct answer. Easier said than done – I kept flying into the trash, and once I flew into the wrong hole! You can shoot at the trash to clear them, but I didn’t do much shooting myself.

The first stage was NOT TOO BAD because there was less trash and I wasn’t timed, but from Stage 2 onwards I had to hurry…if I took too long, an eclipse would happen and I lost a life. Trouble was, I didn’t know about the time limit at first. Worse still, I got the fly and shoot buttons mixed up! Perhaps that was why I had difficulties controlling Blaster’s flight (again!). There came a point when I simply decided to fly straight into a hole as soon as the path was clear, which didn’t always work! I often got hit by the trash and was sent flying back to the ground. That was when I usually ran out of time, and in the end I easily lost all five lives.

To rub the salt into the wound, I later found out that this was the final stage of the whole game – had I finished it, I could have watched Blaster save Spot and set the UFO alight! I was so close! The trash alien gets away, though. Can anyone stop that yellow fellow?


The main obstacle for me was the game controls. Maybe it’s because I barely touched a games console in my entire life – the vast majority of games I played were on a PC, and the same was true when I played Math Blaster as a child, hence I’m used to saving the day with a mouse and keyboard. Also, I kept forgetting which button did what…possibly because I didn’t think of checking the controls option before starting the game!

Also, the Mega Drive version seems quite short compared to the PC version of 1998, which had one more stage in the game, and the trash zapper comes up a second time. Moreover the storyline was expanded on a fair deal – the trash alien is called Gelator, and he is serving his sentence in prison. The ending has its differences, too: if you win the game, Blaster traps Gelator, sending the three-eyed criminal back where he belongs.

Still, I reckon the Mega Drive version is rather nifty game for its time. The graphics were quite impressive, I recognised Blasternaut and co., and I got my dose of Math Blaster nostalgia! Definitely going to play it again.

UPDATE: I played the game a second time, and I finally managed to beat the final stage! Hurray!


Why knot?

As any fule kno…

you can’t tie a knot in a length of string without letting go of the ends.

Let’s be a bit more mathematical: you can’t tie a knot in a loop of string without cutting it. (We can think about the string together with your body as forming a loop, and cutting the loop as letting go of an end.) Continue reading