Funbers 31, 32 and 33

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

31 – Thirty-one

Hands up if you like ice cream? And your favourite brand? I’m not sure I could pick a favourite myself, but Baskin-Robbins is certainly up there. They have a total of 31 flavours of ice cream which means the name of their shops in Japan literally translates as ’31 Ice Cream’. In theory a great idea, but what if they discover a magical new thirty-second flavour…

Aside from frozen goods, thirty-one is also the number of teams in the National Hockey League, with 24 coming from the US and 7 from their Northern neighbours Canada. Each season, the teams battle it out to win the Stanley Cup — the oldest trophy to be awarded in professional sport in North America, and also the one with the infamously large base (see below). Originally, key members of the winning team were engraved on the base, which means its grown a fair few inches over the past 126 years. However, these days the oldest band is removed and replaced with a new one to prevent the trophy from getting any bigger. Less fun no doubt, but perhaps sensible given its already considerable size…

Credit: Michael Miller

Thirty-one is also a Mersenne Prime — the third such one in fact. A Mersenne Prime is a prime number that can be expressed as exactly one less than a power of two: 2^n — 1 for some positive whole number n. To get 31, we take n=5: 2⁵ = 2 x 2 x 2 x 2 x 2 = 32 minus 1 gives 31. A perhaps surprisingly large (and possibly infinite) amount of prime numbers take this form, with the current largest known prime number also being a Mersenne Prime: 2⁸²⁵⁸⁹⁹³³–1 = a number with 24,862,048 digits, aka too many for me to write out here!

32 — Thirty-two

Sticking with maths, thirty-two has the very nice property that it can be written as 1 to the power 1 plus 2 to the power 2 plus three to the power three, or in its neatest form: 1¹ + 2² + 3³ = 32. Here’s a challenge for you: can you work out the next largest number that follows the same pattern?

32 can also be written as 2⁴ + 4² = 32 which makes it a Leyland Number. Any number that can be written using two other numbers x and y, in the pattern x to the power y plus y to the power x, is classified as a Leyland Number. Here, we take x = 2 and y = 4 to get: 2⁴ = 2 x 2 x 2 x 2 = 16, plus 4² = 4 x 4 = 16, giving a total of 16 + 16 = 32. Other Leyland Numbers include: 8, 17, 54, 57 and 100 — I’ll leave it to you to figure out the specific values of x and y needed to satisfy the formula x^y + y^x = Leyland Number for each of the cases above.

Outside of the mathematical world, thirty-two is the number of completed piano sonatas by Ludwig van Beethoven; the number of black (or white) squares, and total number of pieces on a chessboard; the number of teeth generally found in an adult human; and the number of described physical characteristics of the historical Buddha, according to the text of the Pāli Canon in the Theravada Buddhist tradition. It’s a pretty long list, but I think it is best enjoyed in its entirety, so here you go:

Level feet
Thousand-spoked wheel sign on feet
Long, slender fingers
Pliant hands and feet
Toes and fingers finely webbed
Full-sized heels
Arched insteps
Thighs like a royal stag
Hands reaching below the knees
Well-retracted male organ
Height and stretch of arms equal
Every hair-root dark coloured
Body hair graceful and curly
Golden-hued body
Ten-foot aura around him
Soft, smooth skin
Soles, palms, shoulders, and crown of head well-rounded
Area below armpits well-filled
Lion-shaped body
Body erect and upright
Full, round shoulders
Forty teeth
Teeth white, even, and close
Four canine teeth pure white
Jaw like a lion
Saliva that improves the taste of all food
Tongue long and broad
Voice deep and resonant
Eyes deep blue
Eyelashes like a royal bull
White ūrṇā curl that emits light between eyebrows
Fleshy protuberance on the crown of the head

33 — Thirty-three

Let’s start with the bad. Thirty-three is one of the symbols of the Ku Klux Klan, with K being the 11th letter of the alphabet and 3 x 11 or 3 K’s giving 33. It is also believed to be the age of Jesus when he was crucified by the Romans.

On a more positive note, 33 is the longest winning streak ever recorded in NBA history, which was achieved by the Los Angeles Lakers in the 1971–72 season. We also find 33 vertebrae in a normal human spine when the bones that form the coccyx (the tail-like part at the bottom) are counted individually.

Now I don’t say this often — mainly because I think I’m supposed to be impartial when writing these — but this next fun fact is one of my all-time favourites. Long playing records, or LPs as they are more commonly known, are referred to as 33’s in the record industry, because they rotate 33 and a third times per minute when playing on a gramophone. So, next time you see a record player you know what to do…

February 11, 2020 1

Numberphile Pi Million Outtakes

Counting outtakes from the Numberphile Pi Million Subscribers celebration video. Look out for a familiar face/torso at 4:33…

February 6, 2020 0

Funbers Christmas Special

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!!

December 24, 2019 0

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

December 10, 2019 0

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

November 5, 2019 1

Tom Rocks Maths S02 E10

Episode 10 of Tom Rocks Maths on Oxide Radio sees the conclusion of the million-dollar Millennium Problem series with the Hodge Conjecture, a mischievously difficult number puzzle, and the answer to the question on everyone’s lips: how many people have died watching the video of Justin Bieber’s Despacito? Plus, the usual great music from the Prodigy, the Hives and Weezer.

Image credit: Lou Stejskal

October 8, 2019 0

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

September 3, 2019 0

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 10²³.

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

August 6, 2019 0

Tom Rocks Maths: S02 E06

Another fun-filled hour of your favourite two things – maths and rock music – courtesy of Tom Rocks Maths on Oxide Radio. This week I’m joined by two of my students from Teddy Hall, Fran and Tom, who will be explaining their favourite mathematical topics, taking part in a bumper numbers quiz, and sharing some of their music tastes. Plus, the usual dose of Funbers, and excellent music from Panic at the Disco, Sum 41 and Muse.

With thanks to Alice Taylor for production assistance.

July 31, 2019 0