I got this one wrong! How many gifts are there in the 12 Days of Christmas?
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- Опубликовано: 21 дек 2022
- Students like to pose little logic puzzles or math problems to me, and I like to oblige as quickly and cleverly as I can. Unfortunately, in this case, I went to fast, and a student got me.
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She asked me how manu total gifts there were in the 12 Days of Christmas song. Thinking this was a simple sum of consecutive integers situation, I used the nifty little formula n(n+1)/2, which will calculate any sum of consecutive integers from 1 to n. In this case, I plugged in 12, got 78, and answered confidently. And, as it turned out, incorrectly.
At first, I thought the problem might be that I wasn't counting both the partridge AND the pear tree or something, but no, the problem was more subtle than that. In the song, "my true love" gives all the gifts of the previous days every single day. So the sum from 1 to 12 gives the number of gifts for just the twelfth day, but not all the other days before that.
These sums of consecutive integers have another name. They are also called "triangular" numbers because of the way they can be represented as triangular sets of dots. But what we want is actually the next level up: the numbers called the "tetrahedral" numbers. The tetrahedral numbers are the sums of the consecutive triangular numbers. So, for example, the third tetrahedral number would be the sum of the first three triangular numbers.
In this case, we're interested in the 12th tetrahedral number. Fortunately, just like the triangular numbers have a special formula that lets us calculate them directly, so too do the tetrahedral numbers: n(n+1)(n+2)/6. Plugging in 12 yields 364, the actual and correct number of gifts in the 12 Days of Christmas.
#12daysofchristmas #triangularnumbers #tetrahedralnumbers #consecutiveintegers
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Ahhhh, that third three was going to give me a stroke!!!!
I wouldn't have been caught but i didn't know about the last use of Pascal's triangle, so it would've taken a long time.
I would've done all 12 triangular numbers ;)
Great vid. Thanks
ha! well, better to do it the long way than the wrong way, right? thanks for watching!
thats a lot of turtle doves.
4:00 "Drawing is not my strong suit" you sure buddy? That's amazing :)
Thanks for uploading this by the way, got this question in a Christmas quiz we did in a physics lesson and I knew it was the triangular numbers but I couldn't remember how to quickly sum them, ended up having to guess one of the 3 options but I still got it haha
nice! thanks for watching :)
@@polymathematic no problem, also slight mistake the 5th row of pascal's triangle should be 1 4 6 4 1, you wrote 1 4 6 3 1, just confused me a bit with the tetrahedral numbers but I looked it up and i get it now haha, very cool concept - does this generalise to the nth dimension if you were to go down those diagonals further?
I put together a bunch of Christmas trivia questions for my students last Friday. This is one of the one I used.
nice!
And the song makes one for each day of the year. :)
Actually!
I knew what the trick was before I clicked the video, but I didn't know the formula the tetrahedral numbers. Although looking at the formula I'm intrigued. Does that generalize? Are the "k-tetrahedral numbers" (by which I mean the numbers on the kth diagonal of Pascal's triangle. There may be a better name, but I'm not sure if k-dimensional generalizations of tetrahedrons have a name) given by the following formula?
(The nth k-tetrahedral number)=(\prod_{i=0)^(k-1) n+i )/k!
If that's true, I imagine it's already known, but I'm not really sure how to look it up. I'm too tired to even try to prove it myself right now, especially since it would likely require double induction.
Edit: Just tried a few random numbers and rows so I'm convinced that formula is correct. I'm certain it's already known though. Smarter minds than mine have certainly already made the same observation
it's very cool that you noticed that! it's related to the other purpose and name of pascal's triangle: the binomial coefficients. if you're familiar with "choose" notation-so (3,2) is pronounced "3 choose 2" and means the number of ways to select 2 objects out of 3, and also corresponds to the third item on the fourth row of pascal's triangle-the triangular numbers are given by (n+1, 2), the tetrahedral numbers are given by (n+2,3), the next level (sometimes called pentatopic numbers) given by (n+3,4), and so on. i'd be totally lost myself if i ever had to prove it, so i'm content with just noticing it :)
@@polymathematic Ahh, of course, binomial coefficients would do it.
And actually, thinking about it that way, I think the proof is trivial. At least the proof that the numbers on the given diagonal are given by the formula I wrote. That the numbers in the diagonal have the summation property is more interesting though.
I do miss this kind of higher level thinking though. I defended my dissertation a bit over 4 years ago, but I had realized that I would be unhappy trying to make a living out of research, and I haven't really been in math at this level since
Beautiful ❤️
I actually discovered this concept 1 month ago when I was trying to find some relation between cubes and there I found.
The formula I used is to sum upto n is
do the cube of n+1 then subtract n+1 from it and finally divide it by 6.
I actually named it as summation of cumulative frequency 😂😂
seems that there’s a mistake on the pascal triangle
ha! yes, i messed up what should have been a 4 on the fourth row, which ended up causing a problem in the fifth row. i fixed it in the fifth row, but didn't realize it came from the fourth row mistake until later.
Look up the origins of the things you do, words you use, symbols you use etc.
Exodus 23:13, Jeremiah 10:1-5, Galatians 4:10-11
Do not steal terms, festivals, observances, symbols, words, titles, names etc. from pagans. [Deu 12:4, 2 Cor 6:16-17, Exo 20:15, Rom 12:21]
If you want to truly follow the Father, read Scripture and check the origins of things. Believe it or not, but the Father is better than a lot of people think and he can be trusted.
Why is that a 3?!?!!!
You were talking like it was a 4 lol
And it should be