The Maths of Christmas

[afmilacci]

The way I've always worked out triangle numbers is the same but slightly different: (Nsquared + N)/2

[TheSmileyking]

Sequences and series how i love thee.

For you smarty pants out there - prove the formula true for all numbers, say k, with the use of mathematical induction if you like :)

[TastyKlownProduction]

there was another way if you look at the numbers there is a pattern 1,3,6,10 now if you look you'll see that 1+2=3 3+3=6 and 6+4=10 do you see the pattern

[Buggy793]

@TastyKlownProduction

Yeah, in fact that's exactly the idea of triangle numbers! What the formula does is use what's called an arithmetic series. It's a much faster way to calculate them.

[allenbbtechno]

I've known this formula for a long time but I've never known of such a good explanation!

[luckyw4ss4bi]

I felt so proud when i figured this formula out on my own in 5th grade while bored in church. I thought i was the first person to figure it out and i knew i'd get a nobel prize. . .

[illustriouschin]

I do not approve of the term "merrymas" as it is disrespectful to X who was born on this day.

[Lightn0x]

Wow.. nice one.. I've worked out the algebric proof of sum(i)=n(n+1)/2, but this is so creative, I love it :P

[klguitarist9997734]

Isn't this like the Fibbonaci sequence, only it's all of the previous numbers rather than the previous two?

[spookyfbi8]

:O So THAT'S why that formula works! Why do maths teachers & textbooks NEVER explain these things? Drawing out the triangles and making a rectangle makes it so simple and intuitive, and yet most students are just given the formula and told to memorize it, along with a bunch of other formulas. If maths was taught like this, there'd be no need to memorize anything because all the formulas would make SENSE!

[Tyster360]

thumbs up for the mirror paradox in the background.

[JordanIsSamurai]

or you could do 1+12, 2+11 (each pair is the same, and if its uneven the middle one counts as an extra half...ex: 13th day 1+13, 2+12... not counting 7 but then its 6.5x14= 91) ... 6 times 13x6= 78

[HaslamCorp]

are you figuring that they will be too drunk to shake hands again when they leave?

[TFerra]

Gauss 

[avgntgwtg]

78 is it

[GammahooX]

Damn, if i try to post something it just says: "Error, try again". wtf?

[TheFlyingTiger129]

@GammahooX Well, obviously you made it at last... xD

[fredrocks222]

what's your IQ? like 100,000,000,000,000,000,000 or  something close to that?

[BlackCyril666]

Are you a math teacher or professor of some sort? Just curious.

[jackasinory]

really wish id found your channel when i was doing my gcses...might have done better lol.

youre smart AND you make sense, thats rare lol

[dosgav]

i mean 78 lol

[dosgav]

answer 48

[Strijdparel]

Happy New Year and thanks for your vids, they are really nice and i learn a lot.

[heyuder]

easy as pie MAGIC BUNNY, or as easy as pye ??

[pubwebmaster]

A merrymas to you too!

[singingbanana]

Merrymas! :)

[BoBTFish]

Awesome, I was just explaining this to my mum earlier today (I had n(n+!)/2 written on my chalkboard teacup), yet I stupidly didn't explain this perfect example, even though I had been trying to work out the total number of presents (summing the first n triangle numbers, which is apparently n(n+1)(n+2)/6 according to the one that is Google. I've yet to prove this.)

[TheSingingNerd]

Why do Mathematicians confuse Halloween and Christmas? Because OCT 31 = DEC 25

[singingbanana]

Haha, nice one ;) I was going to do that at Halloween but decided to save it. Expect to see it next year.

[pootengs]

@TheSingingNerd i dont get it lol

[TheFlyingTiger129]

@TheSingingNerd lol, that's awesome! xD

[DUBstepgivesmefood]

or the missing number is 18

cause Dolce&Gabbana has a perfume

collection

DG 1, 3, 6, 10, 18 ;D

That's what I had to think about when I saw the commercial ;p

I HAD to think about this video I've seen earlier that day ;p

[lordvalamor]

Nice use of Σ(i, i=1,n) and Pascal's Triangle. I enjoy watching your videos, and I feel this one especially helped me to understand the algorithms more than I already had.

BTW: My goal through college is to be a mathematician. Any tips or other stuff I should know? (I'm a senior in high school, currently taking Calculus I AP.)

[johnlie123]

12*(12+1)/2

[ER3inaction]

calculator 12! right?

[Arlenamadra]

mmhseriously guys there must be some nice men out there i am so lonely

[singingbanana]

I think this might be the wrong place to look.

[vladimir0B]

@Arlenamadra Check out MysteryGuitarMan

[Sniper2053]

78 presents

[EmoEwok666]

wow ur smart

[jkun17]

holy crap! this song has been bugging me since the first time I heard it

I thought I was the only one weirded out by 40 gold rings.

[theboombody]

@jkun17

I don't know, for some reason, I thought that you were only given 5 gold rings on the fifth day, and only given 6 geese on the sixth day. I never thought of it as you getting 5 gold rings every day from the fifth to the final, but I guess the song is worded in a weird way.

[kemazata]

cant you just do 12!

[kyle7447]

My 1st thought was factorial then compinations then permutations. wow i was all over the place haha

[Drag0n247]

also for the handshakes you could just say that, x= n(n-1) divided by 2, and it gives you the same result as the one less triangle number method. so for 51 guests, 51(51-1) = 2550/ 2 = 1275

[zach9500]

thats the same exact formula with the numbers moved around man

[zach9500]

actually my bad its not the same formula but your just switchin n+1 in is formula for n in yours.. your n represents the number of people and his n represents the triangle number

[Drag0n247]

truth and i believe i stated that in my comment

[hi1hi2hi2]

nice

[stricklins]

That was awesome!

[Buzza18]

OMG i love your videos, the only subject i actually liked and did well in at school was maths, which meant, i didnt have many oppertunities to carry on my love of maths much further! so THANK YOU! for all your videos :)

[spartaremix299]

i have to be honest, i cannot see geometry to any relevance to any of careers that are available, or maybe 5 percent maximum, of all careers.

[singingbanana]

It depends what you choose to do and in that respect it is as relevant as any other subject; French, geography, art, music etc.

However in most careers you won't be using pythagoras' theorem everyday, that's not the point. With maths you're showing employers that you are someone who can solve a problem. Not necessarily a maths problems. Employers know that and it's very desirable. Good maths students earn 10% more.

[spartaremix299]

@singingbanana , i understand, but its pretty redundant what i am learning in 10th grade. it gets annoying.

[bbdude77]

Oh my God I love your videos. They're so interesting! :)

[pseudopeople]

Your awesome! much better than my maths teacher!

[MaffiasAngelO88]

Sorry, I spaced, I just didn't follow any of that. It was like I just couldn't absorb the information.

Well maybe next time=)

Merry christmas from Wales!

(Not New Wales 'Australia', Wales Great Britain)

Peace

[jason311094]

wow thanks for the tip! =)

MERRY X'MAS!

[gmsherry1953]

BTW, isn't this the same basic calculation as in the story anecdotally attributed to Gauss as a child? A teacher set the class to adding up the numbers from 1 to 100, and he solved it in seconds? The difference is that in this video, the technique is applied to a variety of problems -- it isn't MERELY adding up numbers in a series.

While researching this (to verify it was Gauss) I found a formula that works for any series where the difference is the same (like 7+11+15+19). Interesting!

[singingbanana]

That's right, I love the Gauss story. In general it's called an arithmetic series.

If you add the first term, a, to the last term, l you get (a+l). If you then add the second term to the second to last term you will get the same value (a+l). Continuing in this way you get n/2 pairs that always add up to (a+l). So the total sum is simply n(a+l)/2.

Triangle numbers are a special case where a=1 and l=n.

[gmsherry1953]

The page where I did my research had a formula based on the first number a, the difference d, and the number of numbers, n (without knowing the last number), where S=na + nd(n-1)/2. It took me an alarming amount of time to realize your formula and that formula are the same. And yours is simpler! Thanks.

[singingbanana]

Yeah :) I put up the simpler version here, but the version with the difference d is possibly more useful. Now look up geometric series, they're cool.

[Slowergrass432]

You're pretty good with maths :D

[gmsherry1953]

I've seen a calculation of the total number of each gift; it's a sort of bell curve, least (12) of the first and last (1 on 12 days and 12 on 1 day), 22 of the 2nd and 11th (2 on 11 days and 11 on 2 days), etc., so I think the 6th and 7th gifts are tied for most (42). But is there an easy way to figure the total number of gifts? The sum of a series of triangle numbers?

[singingbanana]

I thought about that before I dd the video. I don't think there is a nice, clever, way to sum triangle numbers.

[spartaremix299]

in a scientific calculator, there is an exlclamation mark there, so you put that with 12 and im guessing that is the answer.

[singingbanana]

That's called factorial, it's slightly different. 12! means 12x11x10x...x3x2x1.

[spartaremix299]

ahh! right, im sorry, i forgot, i lost my scientific calculator so i could not clarify with my statement.

[generalarthu]

is this related to permutation?

[singingbanana]

A permutation is a way to shuffle n objects (like a pack of cards). The number of ways to do that is 1x2x3x4x...xn (called n factorial and written n!).

It is a different number from the nth triangle number which is 1+2+3+4+...+n (and written T_n).

[Nenhunter]

Well, isn't this related to compinations? i mean it's the same thing with 5C2 ... that's 5!/[(5-2)!2!] = 15 = 4(4+1)/2

[kinsoncheung123]

Merry BIDMAS Jim!

[izn3rdyhomeboy]

my 8th grade algebra mind just exploded O_O

[TyYann]

Great! Now we're ready for factorials!

Merrymas to you too!

[JesseMercArt]

That equation for finding the triangle numbers is almost exactly the same as the equation for finding the area of a triangle: base x height x 1/2. The units would be numbers of gifts.

Isn't Geometry a wonderful class?

God I feel like a nerd. . .

[scotland7yard]

As Eddie Izzard wisely remarked, the only bit I remember of this song is the FIVE GOLD RINGS !!!!!!!!

[singingbanana]

heehee

By the way, wikipedia says the rings are also birds.

[scotland7yard]

indeed it does.

and nowadays i can say wikipedia can be trusted even more than its own sources

[TheHungarianCuber]

78 is the number

[08AOvechkin08]

do a segment on gauss!

[singingbanana]

I love the Gauss story.

[dungeonmunky]

Woah ho ho. I love learning different ways of thinking about things I already know, namely the sum from i=1 to n of i. I haven't heard the term triangle number before. Neat video!

[123Griffin456]

you just made maths awesome again

[link211211]

plz talk about limits, rolle and bolzano!!!

[singingbanana]

Noted.

[okmasko]

Jim, you are really my favourite subscription :) Thank you.

[reclusivesage]

I wonder which gift you would have the most of at the end of the 12 days.

[singingbanana]

Ah... I could have done that. Decided not to for this video. Easy enough to add up, maybe someone would like to try.

[reclusivesage]

184 birds...that's at least 183 too many.

[sebbbaastian]

You should move to Sweden and be my math teacher.

[computerman91]

nice

[reecehalo99]

lol, i wish you were my math teacher

[jcyr84]

isnt this pascale's triangle??

[MrCrownKing]

cool back ground

[polywoly15]

i don't like how you left out "christ" in christmas =/. anyways great vid as always.

[misssoftglitterkid]

you make math so cute

[loukas1997]

great video

[phanatadashi]

wow i just learned this in math but you explained it much better

[Seigge]

wow, math is amazing , YOU'RE amazing!

[choco33333333]

im beginning to like math!

[nikolka1992]

That's why I love math.

[Duckweed73]

I like this vid!

[MagicRevealer09]

You can fit maths into anything! 5 star!