A lot of people write one-tenth as 0.1, since the decimal can be overlooked and that creates confusion. That would make the claim work, but of course you don’t have to write the leading zero.
I don’t get it exactly but I found this.
http://www.applet-magic.com/Digitsum00.htm
My contribution:
Harper’s index for this month put it at $1.6 bn.
So what is the relationship between integer cubes and modulo 9 arithmetic?
Cite says:
The Equivalence of Digit Sum Arithmetic
with Modular 9 Arithmetic
So: Cubes->Digit Sums->Modulo 9.
I don’t understand how cubes generate those consistent digit sums. But this shows how digit sums convert to Modulo 9. And that’s what I was referring to (below).
I’m not an authority on… well, anything, but I don’t believe a guitarist can play rhythm and lead simultaneously — except in extremely unnatural circumstances. A guitarist can certainly play rhythm and lead during the same song, switching from one to the other. And a guitarist can certainly play patterns that contain prominent rhythmic elements and prominent melodic elements.
But for a guitarist to play normal rock rhythm guitar patterns while also playing normal rock lead guitar fills, intros, and solos, will require tape loops, or (again) “unnatural circumstances.” With some practice, a guitarist could play a rhythm track on the bottom three or four strings and hammer out a lead line on the top three or four strings; that’s something any skilled musician could do, but both parts will be more limited than you can get by just having a second guitarist.
As for singing on top of simultaneous rhythm and lead playing, lead guitar licks typically go in the spaces between the vocals — rather than occurring simultaneously with the vocals.
TLDR: Saying that Mr. Kath could do this unusual 3-things-at-once feat is more of an affectionate compliment than a description of a rare ability.
I know, right?
Ever see when people have the bridge of nerves between the two halves of the brain severed? Split brain research is pretty wild.
Maybe Kath was a freak of nature. Here’s the primary source (CNN) from which Wikipedia took the info:
https://edition.cnn.com/2016/12/16/us/history-of-chicago-guitarist-terry-kath/index.html
I guess I shouldn’t have asked about the relationship, which is just an arrow. I’m looking for an explanation of why integer cubes do this.
Let me give it a try, different from the explanation linked above. When you write a number in decimal notation as, say, abcde, that stands for the sum
a104 + b103 + c102 + d10 + e.
Now think about how much larger this is than a multiple of 9. Each 10n is just 1 larger than a multiple of 9, since 99…9 (n 9s) is just below it. It follows that we can replace each 10n with 1 and say that abcde is a+b+c+d+e larger than a multiple of 9. If that sum isn’t a single digit, do it again: The sum of its digits will be the the same amount above a multiple of 9 as it was. Keep doing this until you get down to a single digit, and that’s the remainder on dividing the original abcde by 9. (One caveat: If your original number is divisible by 9, you’ll get 9 as the digit sum rather than 0, but that’s a relatively harmless detail, though a good part of what the earlier link is about.)
There’s nothing special about 9 or cubing for what follows: For any integers n and p, write n % p for the remainder on dividing n by p. Then
n3 % p = (n % p)3 % p.
That is, to get the remainder on dividing n3 by p, you can first find the remainder on dividing n by p and then cubing (and then finding the remainder again, as what you get will likely be larger than p). The upshot is that, if you want to know what remainders you get on cubing numbers and then dividing by 9, you’ll get exactly the same remainders that you get for the sequence of numbers 0 through 8. When you get to 9 through 17, you’ll get the same sequence repeated, and so on.
I think you must be very good at math but your understanding of the term ‘dim-witted’ seems to be lacking. If no one else does, I’ll open a GQ thread about it later.
Let’s start with the fact that is an integer is divisible by 3, that is 3k, then its cube is divisible by 9 since (3k)^3 = 27k^3 = 9(3K^3) or 0 mod 9.
For the only other two possibilities, let A = 3k+1 and B = 3k+2
By using the identity (x+y)^3 = x^3 + 3[(x^2)(y)] + 3[(x)(y^2)] + y^3
A^3 = (3k+1)^3 = 27k^3 + 3(9k^2) + 3(3k) + 1 = 9(3k^2 + 3k + k) + 1 or 1 mod 9
B^3 = (3k+ 2)^3 = 27k^3 + 3(18k^2) + 3(12k) + 8 = 9(3k^2 + 6K + 4k) + 8 or 8 mod 9
Sorry for all of the carets. Anyone know how we would do superscripts and subscripts?
Train and Tram doors is a thing, apparently. And elevator doors too! There are more.
(Sorry to interrupt the math class.)
Ok, big help, that’s the connection that I was looking for. Have to look at this more for a while though, those big integers greater than 1 always confuse me.
It’s kind of like telling time - There are 24 hours in a day, but the clock face for each hour is the same, the images of the minutes are congruent. So, for the minute hand, 3:15 looks the same as 5:15 on the clock face because hour group 3 is the same (congruent with) as hour group 5. We are observing that the digit sums of integer cubes form congruent groups.
I’m not sure of the mechanics, but it helps to write a matrix of the sequential integers. You have three columns with 1,2,3 across the top. Row 2 is 4,5,6 and row 3 is 7,8,9 etc. Notice that in column 3 all the numbers are divisible by 3 so cubing them will produce a 9 and all numbers divisible by 9 and some divisible by 3 will have the digit sum of 9. Also every other number in column 3 is the result of 6xN and primes fall into the group 6xN±1. So there are no prime numbers in column 3. They are in column 1 or 2. There are no primes with the digit sum of 9.
Notice that 6x5=30 that yields the prime 29 in column 2 and the prime 31 in column 1. Their cubes are 24389 (8) and 29791 (1). I don’t immediately see what determines that, but maybe one of the mathematicians here will. Or perhaps, if we stare at it long enough the reason will jump out. That’s what Newton did.
Reading up after posting in the thread about bullying Major Frank Burns in TV’s MASH, I consulted this article:
Frodo went to live with his rich (but unpretentious) uncle Bilbo Baggins in the mansion at Bag End, always coveted by the snooty Camellia (nee Sackville) and her daughter Lobelia.
If it was J. K. Rowling writing that, people would have said it was a cheap joke…
Nitpick: although all Concordes were grounded immediately following the (one and only) fatal crash, most were upgraded to prevent a repeat of the problem (which was mainly caused by a bit of metal on the runway that shouldn’t have been there) and returned to service. However, you are right in that it was the effective end of supersonic passenger aviation (at least, to date), as passenger numbers never quite recovered, although 9/11 and modern technology also played big roles in that. Concorde’s last commercial flight was in 2003, which was largely an alcohol-soaked celebrity bunfight, and its last flight ever came shortly after, as it flew over my hometown (and its place of manufacture) before landing at the museum where I have since seen it. Wish I could have flown on it.
Excellent username/post combo (yes I know it’s not quite technically perfect, but still).
Thank you - this was the most helpful explanation for me.
Some of the language in the podcast is very rough, so there’s my NSFW warning.
Chuck Negron (one of the lead singers of Three Dog Night) says he could have been killed in the Wonderland murders—his wife found the victims. Listen at about 25:15.
Also, he had so much sex and drugs that his penis exploded and he needed stitches in it. Jump to about 30:00.
I have to send this to my DIL. She really decorates for Halloween.
TIL that chickens eat mice. They can, in fact, be excellent mousers.
I knew they ate bugs and worms, but mice? MICE?
They really are little dinosaurs. And I am completely, totally boggled.