A very fun Christmas treat for you all as I team up with my good friend Bobby Seagull for the Funbers Xmas Special – expect fun facts, lots of numbers, and more birds than anyone thought possible… Happy Holidays!!
Tom Rocks Maths S2 E12
Season 2 comes to a close with stories from my (rather eventful) trip to China, a new video series with BBC Maths Guru Bobby Seagull, and the number of calories needed by a Charizard per day to survive. That’s all on top of the usual puzzle and fun facts about the numbers 0 and 1. Plus, music from the Red Hot Chili Peppers, System of a Down, and Limp Bizkit. This is maths, but not as you know it…
Tracklist:
- 00:00 Opening
- 00:13 Bowling for Soup – Normal Chicks
- 03:25 Limp Bizkit – My Generation
- 07:05 Red Hot Chili Peppers – The Adventures of Rain Dance Maggie
- 11:44 News
- 18:00 Enter Shikari – Arguing with Thermometers
- 21:22 Puzzle
- 24:09 System of a Down – Shimmy
- 25:59 Atreyu – You Gave Love a Bad Name
- 29:22 Pokemaths: How many calories does a Charizard need per day?
- 38:13 Midtown – Get it Together
- 41:30 Billy Talent – Nothing to Lose
- 45:02 Funbers 0 and 1
- 51:36 The Story So Far – Right Here
- 54:02 Puzzle Solution and Close
Funbers 28, 29 and 30
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
Funbers 25, 26 and 27
Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…
25 – TWENTY-FIVE
You probably know 25 as five squared, 5 x 5 = 25, but I bet you didn’t realise that it’s also the sum of the first five odd numbers: 1 + 3 + 5 + 7 + 9 = 25. It also crops up a lot in Pythagoras’ Theorem (yes, him again — see Funbers root 2) because it’s the smallest square that’s also the sum of another two square numbers: 25 = 3² + 4². Since Pythagoras’ Theorem says that a² + b² = c², we have the exact result with whole numbers (integers) for a = 3, b = 4 and c = 5. A solution such as this, where all of the numbers are integers, is called a Pythagorean Triple.
Looking beyond the maths, most videos are recorded at a frame rate of 25 per second as the PAL video standard – other options are available, but twenty-five does an excellent job of tricking the human brain into seeing a moving picture where in fact only a series of still images are being shown. Less than 25 and we might start to notice the ‘jumps’ between frames, and for more than 25 we’ll need a lot more data to record and store the footage.
Twenty-five is also the average percentage of DNA overlap between yourself and your grandparent, grand-child, aunt, uncle, nephew, half-sibling, double cousin (when siblings from one family have children with siblings from another), or identical twin cousin (if one of your parents is an identical twin and their twin has a child). Oh, and apparently a ‘pony’ is British slang for £25 – news to me…
26 – TWENTY-SIX
With twenty-five being a square number, and (spoiler alert) twenty-seven being a cube number, twenty-six is uniquely placed as the only whole number that’s exactly one greater than a square (5² + 1) and one less than a cube (3³ – 1). Talk about niche. And then there’s the fantastically named rhombicuboctahedron — a shape with 26 faces, made up of squares and triangles. Can you spot how many of each in the figure below?
Twenty-six also gives the number of complete miles in a marathon (26 miles and 385 yards to be exact), the number of letters in the Latin alphabet, and the age at which males can no longer be drafted in the United States. The draft has been used five times throughout history: the American Revolution, the American Civil War, World War 1, World War 2 and the Cold War (including Korea and Vietnam). Let’s hope it never has to be used again.
27 – TWENTY-SEVEN
Now this one’s a real doozy: 27% of our universe is made up of “dark matter” – matter that has mass but is also completely invisible and doesn’t interact with itself or regular matter. The rest of the universe consists of 5% regular matter (the stuff we know about), and the other 68% is completely unknown. Something, something, dark energy…
Sticking with scary thoughts, in Stephen King’s novel ‘It’ (great film by the way) the creature returns to the town of Derry every 27 years, which also happens to be exactly the right amount of time for a new-born baby to join the 27 Club — a term used to refer to popular musicians who have died at the age of 27. Current members include Jimi Hendrix, Kurt Cobain and Amy Winehouse amongst many, many more. We also have 27 books in the New Testament and 27 bones in the human hand.
Ending with some maths — what else — twenty-seven is the only positive whole number that is exactly three times the sum of its digits: 2 + 7 = 9 and 9 x 3 = 27. It’s also a perfect cube, 33 = 3 x 3 x 3 = 27, and it’s equal to the sum of the digits from two to seven, 2 + 3 + 4 + 5 + 6 + 7 = 27. But, leaving the best until last, if you label the decimal places of the number pi, starting from 0, then the 27th and 28th digits read 27. It may seem like magic but it’s actually one of a few ‘self-locating strings’ in the number. The others being 6, 13598, 43611, 24643510, and no doubt many more yet to be discovered. That can be your homework…
π = 3.141592653589793238462643383279…
Image credit: Jonathan Kis-Lev
Funbers 22, 23 and 24
Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…
22 – TWENTY-TWO
Coming in hot, 22 happens to be one of my favourite numbers – if you divide it by 7 you get about 3.142, which is a handy way of getting close to pi without having to remember all the digits! Then of course there’s Joseph Heller’s famous novel Catch-22. In the book, Catch-22 is the Air Force policy which says that bomber pilots can only stop flying planes if they are declared insane. But like the name suggests, there’s a catch. Catch-22 says that asking for a mental evaluation to get declared insane is proof that you aren’t in fact insane. So technically, there’s a way to get out of flying more bombing runs… but if you try it, you get sent right back out in the next plane!
Twenty-two also pops up in the kitchen. Normally, if you are slicing a pizza using 6 cuts, you’d do it neatly and end up with 12 even slices – much like the numbers on a clock face. But if you were a lazy pizza chef and just sliced randomly, you could end up cutting slices in half and ending up with more pieces. And it turns out, the most pieces you can end up with after 6 cuts is, you guessed it, 22!
On a darker note, 22 was also the lucky number of the Haitian voodoo dictator Francois “Papa Doc” Duvalier. Papa Doc started studying voodoo folklore to spread rumours that he had supernatural powers, which let him rule through fear. But eventually, he started believing the rumours himself. He would only go outside his palace on the 22nd of the month, because he thought he was guarded by voodoo spirits on that lucky day. He even claimed to have killed JFK, whose assassination was on the 22nd of November 1963, supposedly by stabbing a voodoo doll of him 2222 times that morning…
23 – TWENTY-THREE
For 23 we’re going back to maths, and specifically prime numbers. A prime number remember, is one that can only be divided by itself and one without giving any remainder. Twenty-three has the unique property of being the smallest prime number which is not a ‘twin prime’ – that is a prime number which does not have another one within two spaces of it on the number-line. For example, 3, 5 and 7 are all close friends, while 11 and 13 go together. 17 is next to 19, but the nearest prime number to 23 is either four places below at 19, or six places above at 29, making it the smallest prime number to not have the ‘twin’ property.
Twenty-three is also big for birthdays. Not because the age of 23 is particularly special (although being the age mentioned in my favourite song – Blink 182’s ‘what’s my age again?’ – I do have a soft spot for it), but because of its appearance in the ‘Birthday Paradox’. The complete explanation is a little too long for Funbers, but in short it says that if you choose 23 people at random and put them in a room together, there is a greater than 50% chance that 2 of them share the same birthday. If that sounds too crazy to believe, check out a full explanation here from one of my students who applied it to the 23-man England squad for the 2018 Football World Cup. Now to enjoy some classic pop punk: “Nobody likes you when you’re 23…”
24 – TWENTY-FOUR
Who remembers Avogadro’s constant for the number of atoms contained in one mole of a substance from high school Chemistry? No, me neither. But, a great way to approximate it is using 24 factorial – or 24! in mathematical notation. The factorial function (or exclamation mark) tells you to multiply all of the numbers less than 24 together. So, 24! is equal to 24 x 23 x 22 x 21 x 20 x 19 x … x 2 x 1, also known as an incredibly large number. It’s about 3% larger than Avogadro’s constant, but certainly easier than remembering 6.02214076 x 10^{23}.
Twenty-four also represents the number of carats in pure gold, the number of letters in the Greek alphabet (ancient and modern) and the number of points on a backgammon board. Mathematically, 24 is the smallest number with exactly 8 numbers that divide it – can you name them? And, it’s equal to exactly 4 factorial: 4! = 4 x 3 x 2 x 1 = 24. Last but not least, where would we be without the 24 hour day – or to be precise 24 hours plus or minus a few milliseconds to be completely exact…
Day length | |
Yesterday | 24 hours -0.46 ms |
Today | 24 hours -0.39 ms |
Tomorrow | 24 hours -0.35 ms |
Shortest 2019 | 24 hours -0.95 ms |
Longest 2019 | 24 hours +1.67 ms |
Last Year Average | 24 hours +0.69 ms |