Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…

28 — Twenty-eight

28 has the infamy of being the second perfect number. This may sound like it came in second place in some kind of ‘best number competition’, but in fact a perfect number is one where all of the numbers that divide it, perfectly add up to give the number itself: 1 + 2 + 4 + 7 + 14 = 28. We first talked about perfect numbers back in Funbers 4.6692…, 5 and 6, which gives you a pretty big hint as to what the first perfect number might be…

Twenty-eight is also a triangular number. Building equilateral triangles using only dots gives rise to a sequence of numbers, each of which is called a triangular number. We start with a single dot, then we add a row of two dots below to make a total of 3, then we add a third row with three dots to give a total of 6, etc. etc. (see below for examples). For a triangle with seven rows the total number of dots will be 28. If you feeling brave, try to work out the general formula for the number of dots in a triangle with n rows.

Credit: Melchoir

So far I think we can say 28 is doing pretty well being both a perfect number and a triangular number, but it doesn’t stop there. Twenty-eight is also a magic number (yes, really). Magic numbers are a concept in nuclear physics which correspond to the total number of protons and neutrons required to completely fill a shell within an atom. There are seven magic numbers known so far: 2, 8, 20, 28, 50, 82, and 126 with at least another eight predicted by the theory.

29 — Twenty-nine

Here’s a fun challenge for you: using the numbers 1, 2, 3, 4 only once, along with the four basic operations of addition, subtraction, multiplication and division, can you make a total of 29? What about all positive numbers less than 29?

It turns out that 29 is in fact the smallest positive number that CANNOT be made using the method described above (it’s Funbers, you should have known there was a twist). In other claims to fame, it takes Saturn just over 29 years to orbit the sun, there are 29 states in India, and 29 Knuts make one Sickle in the currency of the wizarding world of Harry Potter. The real question is how many Sickles make a Galleon?

Credit: SunOfErat

30 — Thirty

As we enter the fourth decade of Funbers, let’s look back at some of the interesting numbers we’ve met so far… 1, 4, 9, 16 — what do they all have in common? Adding up the first four square numbers gives exactly 30: 1² + 2² + 3² + 4² = 30. This property makes it a square pyramidal number or a cannonball number. The latter name comes from the fact that a square pyramid can be built from exactly 30 cannonballs — instructions below if you want to try it out for yourself, though I recommend using something lighter and easier to obtain than medieval ammunition.

If you’re lucky enough to reach 30 years of marriage, you celebrate the Pearl Wedding Anniversary where, as the name might suggest, you traditionally receive a gift of pearls, although the ‘modern’ list published by the Chicago Public Library suggests a gift of diamond instead. Either way, sign me up. There are in fact suggested gifts for most wedding anniversaries — too many for me to include them all — so here’s a selection of some of my favourites. I’ll let you figure out which is the traditional gift and which is the ‘modern’ one…

1st — Cotton or a clock

3rd — Leather or glass

7th — Wool or a pen and pencil set

8th — Salt or linens

24th — Opal or musical instruments

85th — Wine or your birthstone

90th — Stone or engraved marble

Finally, let’s end with the magical and mysterious date of February 30th. It of course does not occur on the Gregorian calendar, where February contains 28 days in a typical year and 29 days during a leap year, and so is often used as a sarcastic date to refer to something that will never happen or will never be done. That is, unless you happened to be living in Sweden during the year 1712. Instead of changing from the old Julian calendar to the new Gregorian calendar by skipping forward 11 days as had been done in other countries, Sweden decided to do things differently. Their plan was to omit all of the leap days from 1700 to 1740, which would in theory have the same result, just over a longer time period.

There were 11 leap years during this timeframe (1700, 1704, …, 1740) and so this approach would have indeed worked, were it not for the Great Northern War. The war began in late 1700 and lasted for over 20 years, which unfortunately caused Sweden to ‘ forget’ to omit the leap days in 1704 and 1708, leaving them on neither the Julian or Gregorian calendars. To avoid confusion (and likely further forgetfulness) they restored the old Julian calendar in 1712 with the addition of the magical day of February 30th (visible in the image below). Which reminds me, I must let Taylor Swift know I can’t make our dinner date on February 30th…

Cover image credit: Lozikiki

4.6692… – FEIGENBAUM’S CONSTANT

A new addition to the list of important mathematical numbers, Feigenbaum’s constant was only discovered in 1978 through the study of chaotic systems. Chaos in the mathematical sense is pretty much what you might expect – it describes something that is completely unpredictable. My favourite way to think of it is using magnets. If you have two magnets each with a North and South pole (those funky blue and red ones from school) and hold opposite ends near to each other, then they are attracted and will quickly move together. This is a nice stable system – we know what will happen every time. And if you slightly adjust how far apart you put the magnets, or at what angle you hold them near to each other, it won’t make a difference; they will still join up. Now, what happens if you try and hold two of the same ends near to each other? Well, for starters they’re now going to repel. But, what’s important here is that they move around unpredictably. Try it. If you force them towards each other they push back and can end up moving sideways, up, down, pretty much in any direction. And if you try to repeat the same movement you can’t. That’s chaos.

So, what does Feigenbaum’s constant have to do with all of this? The answer lies in fractals — a repeating pattern that continues to look the same despite zooming in further and further (see image below). The rate at which the image zooms in is Feigenbaum’s constant — cool, huh?

Feigenbaum’s constant is important because it describes the rate at which the simplest mathematical systems (called one-dimensional maps) descend into chaos. The really awesome thing is that you can create all kinds of different systems (following a few basic rules) and they will all go into chaos at the same rate given by this exact constant.

5 – FIVE

Apart from being a hit(?) boyband from the 90’s, five is also a number. It has the fun quirk of being the fifth number in the Fibonacci sequence: 1, 1, 2, 3, 5… where the next number is the sum of the two before it. It’s also the number of human senses (or at least the main ones) and the number of rings in the Olympic symbol. My favourite use of five, however, has to be the 5 Platonic Solids. These are the most, regular, symmetrical and beautiful shapes in all of maths – some people would say that they are so beautiful that they should be permanently inked onto your body… who are these crazy people? Hint: it’s me.

To be a Platonic Solid you need to be 3D, with each face the same shape, and at each corner the same number of faces must join together. Take the simplest example: the cube. We have 6 faces that are all squares and at each corner 3 squares join together. The smallest Platonic Solid is the Tetrahedron or triangle-based pyramid which has 4 faces that are all triangles. After the square comes the Octahedron which has 8 faces that are all triangles – basically two square-based pyramids stuck together so that the square face is inside the solid. The last two are the Dodecahedron with 12 pentagon-shaped sides and the Icosahedron which has 20 triangles all stuck together. These are the only 5 shapes that satisfy the very simple set of rules and they appear everywhere in nature, from the shape of viruses to the structure of molecules. In short, they’re grrrrreat (credit to Tony the Tiger).

6 – SIX

The number of impossible things that the Queen from Alice in Wonderland believes before breakfast and also the number of legs on every insect on earth. Insects are in fact the largest group of species we have on the planet, and even outnumber all of the other species combined… We also have 6 Quarks, which as well as being some of the fundamental particles that make up our universe, have the fantastic names of up, down, bottom, top, charm and strange. Sounds like a fun weekend…

Mathematically, six is what we call a perfect number: all of its factors (the numbers that divide it) add up to give six 1 + 2 + 3 = 6. It also happens to be the only number in existence where not only do all of its factors add up to give the number, they also all multiply together to give the number too: 1 x 2 x 3 = 6. Perfect numbers (the ones that equal the sum of their factors) are a little more common, the next three are: 28, 496, 8128. There are more of course, but they get a little tricky to find. Be my guest and see if you can work them out… sounds like another fun weekend to me.

You can find all of the funbers articles here and all of the episodes from the series with BBC Radio Cambridgeshire and BBC Radio Oxford here.

Tag: perfect number

Funbers 28, 29 and 30

28 — Twenty-eight

29 — Twenty-nine

30 — Thirty

Funbers 6

Funbers 4.6692… 5 and 6