B-b-but did you also calculate how much taller the cat is than the turtle?
― A Kestrel for a Neve (James Redd and the Blecchs), Thursday, 5 January 2023 14:39 (one year ago) link
anywhere between 1.15x and infinitely...
― ledge, Thursday, 5 January 2023 14:59 (one year ago) link
yes i know that too
― mark s, Thursday, 5 January 2023 15:06 (one year ago) link
tab + tur - cat = 130, tab - tur + cat = 170, just add them together to get tab + tab = 300
― koogs, Thursday, 5 January 2023 15:38 (one year ago) link
hence cat - tur = 20, however i do not believe we can calculate cat or tur in themselves despite knowing the diff between them
― mark s, Thursday, 5 January 2023 15:51 (one year ago) link
i assumed tur=0 to (marginally) simplify things
― ledge, Thursday, 5 January 2023 15:52 (one year ago) link
MIND YOUR DECISIONS ILXOR LEDGE
― mark s, Thursday, 5 January 2023 15:55 (one year ago) link
but both the creature heights disappear when you add the two, no need to assume anything!
― koogs, Thursday, 5 January 2023 16:22 (one year ago) link
i know. nevertheless, i like assuming.
― ledge, Thursday, 5 January 2023 16:24 (one year ago) link
I used the sin double angle formula.
― A Kestrel for a Neve (James Redd and the Blecchs), Thursday, 5 January 2023 16:26 (one year ago) link
No I didn’t but it is vaguely related
i do not believe we can calculate cat or tur in themselves despite knowing the diff between them
― mark s
yes, more unknowns than equations
even more unknowns: how many kids (out of all the kids in china) are doing this problem, how many do it correctly, do they do it with help, are they expected to get it right, why do we even care
― the late great, Thursday, 5 January 2023 16:31 (one year ago) link
Here's a question for y'all: what are the maximum heights of the cat and the turtle?
― A Kestrel for a Neve (James Redd and the Blecchs), Thursday, 5 January 2023 16:48 (one year ago) link
also, how is that table still standing with that cat sat on the edge like it is?
― koogs, Thursday, 5 January 2023 16:57 (one year ago) link
table is bolted to the floor, next question
― the late great, Thursday, 5 January 2023 17:04 (one year ago) link
maximum height of the cat is the height of the table, no? what's interesting is that as you make one animal taller, the other gets taller with it. that seem counterintuitive to me but that intuition is probably just based on my accumulated experience of (superficially) similar problems
― the late great, Thursday, 5 January 2023 17:10 (one year ago) link
the picture breaks if the cat gets bigger than the table but I'm not sure the maths does.
― koogs, Thursday, 5 January 2023 20:02 (one year ago) link
Right. Turtle will eventually break the table too.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Thursday, 5 January 2023 20:14 (one year ago) link
have only just noticed that this is the average of the two values. is that always true?
― koogs, Thursday, 5 January 2023 20:19 (one year ago) link
yeah, trivial
― koogs, Thursday, 5 January 2023 20:21 (one year ago) link
yes bcz of yr post at 16.22 london time
― mark s, Thursday, 5 January 2023 20:23 (one year ago) link
https://www.youtube.com/watch?v=tS8oyl1gygs
― Farewell to Evening in Paradise (James Redd and the Blecchs), Thursday, 5 January 2023 20:53 (one year ago) link
Slight variation on the problem: instead of limiting ourselves to 2 measurements and 2 animals, imagine we could do a million measurements. For each measurement we select at random two animals from the total animal population of earth. We place one animal on the floor and one on the table (assume table is of infinite strength), and measure the distance between the top of the floor animal to the top of the table animal (which could be a positive or negative number). Given measurements M1 through M1000000, what is our best estimate of the actual height of the table?
― o. nate, Friday, 6 January 2023 17:51 (one year ago) link
51"
― G. D’Arcy Cheesewright (silby), Friday, 6 January 2023 17:52 (one year ago) link
Bear in mind that if we select 2 million animals at random from total earth animal population, its likely that most or all will be bacteria and hence of negligible height.
― o. nate, Friday, 6 January 2023 18:48 (one year ago) link
bacteria aren't animals!
― G. D’Arcy Cheesewright (silby), Friday, 6 January 2023 18:57 (one year ago) link
You're right. Please disregard my previous statement. It seems the most common animal will be an insect, whose height may or may not introduce significant error in our measurement, depending on the height of the table.
― o. nate, Friday, 6 January 2023 20:01 (one year ago) link
Come now Let us be crooked but never common.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Friday, 6 January 2023 20:05 (one year ago) link
Challenging myself to figure out why this works:
https://pbs.twimg.com/media/FmBgdAxWAAA_O_h?format=jpg&name=small
― o. nate, Monday, 9 January 2023 19:17 (one year ago) link
Happy to explain if you don't figure it out
― Guayaquil (eephus!), Monday, 9 January 2023 20:03 (one year ago) link
Oh, I'll let it percolate in the back of my mind for at least a day or two before relenting and looking for hints. Interestingly this trick doesn't preserve the remainder of the number after division by seven in the general case. Only in the case of remainder zero. So its not a perfect modular algorithm.
― o. nate, Monday, 9 January 2023 20:20 (one year ago) link
Took me a few minutes of percolating but it makes perfect sense now.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Monday, 9 January 2023 20:28 (one year ago) link
Thought of two ways to do it. One is more obvious and clear, one is slightly fancier but more interesting. They both amount to the same thing anyway.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Monday, 9 January 2023 20:40 (one year ago) link
Can’t believe I never came across that before. I did spend time long ago thinking about why the decimal representation of one seventh is what it is though.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Monday, 9 January 2023 20:58 (one year ago) link
Not much percolating in my brain yet, except a vague idea that the equation 5 * 10 - 1 = 7 ^ 2 is somehow involved.
― o. nate, Monday, 9 January 2023 22:44 (one year ago) link
facts
― Guayaquil (eephus!), Monday, 9 January 2023 22:44 (one year ago) link
as a physical sciences person i suck at number theory. here’s how i can prove it for numbers between 100 and 1000, no idea how to generalize to the result. pretty sure a slick method would use mod but as i suck at number theory idk how to do that
suppose you have a three digit integer with digits p, q and r which can be written 100p + 10q + r
using chika’s method we drop the last digit r and divide by 10, then add 5r. assuming this new number is divisible by 7, we can write
10p + q + 5r = 7m, where m is some rando integer
multiply both sides by 10 to get
100p + 10q + 50r = 70m
to reconstruct our original integer, we subtract 49r from both sides to get
100p + 10q + r = 70m - 49r
since 70 and 49 divide by 7 (and m and r are integers) we’ve proven 100p + 10q + r is divisible by 7
some distance still to go before i’m as slick as a 12 year old math whiz
― the late great, Tuesday, 10 January 2023 00:51 (one year ago) link
actually maybe it’s not as hard as all that - with a four digit number i think you just get 700m on the other side, 7000m with five digits, and so on?
sadly i think you always get a -49r and never a 490 ;_;
― the late great, Tuesday, 10 January 2023 01:03 (one year ago) link
er no, scratch that, i think you always have 70m, just bigger and bigger junk in front of the 50r to keep track of
― the late great, Tuesday, 10 January 2023 01:10 (one year ago) link
One way:Think instead of just peeling off the last digit, multiplying it by 50 and adding it back to the rest. This will just be ten times the number in the suggested trick and is divisible by seven if and only if that number is. If the original pieces were p and q mod 7, the new number mod 7 is p + 50q = p + 49q + q which is again p + q mod 7, so the operation does not change divisibility by 7
― Farewell to Evening in Paradise (James Redd and the Blecchs), Tuesday, 10 January 2023 01:11 (one year ago) link
yeah see that’s slick, i understand it but i just can’t generate math like that
― the late great, Tuesday, 10 January 2023 01:17 (one year ago) link
Other way: Do the actual trick as specified, divide one part by ten and multiply the other by five, so p/10 + 5q. What is integer division by 10 mod 7? Well, 10= 7 + 3. What is the multiplicative inverse of 3 mod 7? 3 * 5 = 15 which is 1 mod 7 so 5 is that inverse. In this case the new number is 5p + 5q mod 7 which again is divisible by seven if and only if the original number is
― Farewell to Evening in Paradise (James Redd and the Blecchs), Tuesday, 10 January 2023 01:20 (one year ago) link
Basically this is a further generalization of the principle that makes the divisibility tests for 3,9 and 11 work, except none of us have ever seen it before, unless someone has, please speak up if so.
― Farewell to Evening in Paradise (James Redd and the Blecchs), Tuesday, 10 January 2023 01:23 (one year ago) link
I don't think I follow your second explanation there, James, but the first one is basically what popped into my head as I was walking my dog this evening.
― o. nate, Tuesday, 10 January 2023 03:14 (one year ago) link
Here's how I phrased it to myself in order to convince myself that it has to work:
The algorithm as stated (taking the last digit, multiplying it by 5, adding it to the remaining number, and then dividing by 10) is equivalent to another hypothetical algorithm (finding the multiple of 49 which when added to the original number produces a multiple of 10, and then dividing out that factor of 10). Because neither of the steps in the hypothetical algorithm change the property of being divisible by 10, neither does the original algorithm.
― o. nate, Tuesday, 10 January 2023 03:18 (one year ago) link
Sorry, that last sentence should have read "property of being divisible by 7".
― o. nate, Tuesday, 10 January 2023 03:22 (one year ago) link
Has to do with groups and rings.https://www.maths.usyd.edu.au/u/UG/SM/MATH3062/r/lect1.pdf
― Farewell to Evening in Paradise (James Redd and the Blecchs), Tuesday, 10 January 2023 03:23 (one year ago) link
https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/06/MA111_Z6and-Z7.pdf
― Farewell to Evening in Paradise (James Redd and the Blecchs), Tuesday, 10 January 2023 03:25 (one year ago) link
Yeah, I recognized it had something to do with theories of modular arithmetic, but was not well versed enough in that subject to follow the condensed explanation.
― o. nate, Tuesday, 10 January 2023 03:25 (one year ago) link
I think my garbled my explanation, but hopefully the gist comes through.
― o. nate, Tuesday, 10 January 2023 03:29 (one year ago) link