Oh wow, I really like the idea of turning gerrymandering into a puzzle! It reminds of puzzles like Nonograms, KenKen, and so on. And, it can teach you a thing or two about the real world's districting-patterns!
yeah, I love those types of puzzles! especially nonograms, those sorts of things are what especially inspired me to make this into a type of puzzle! it was just such a funny idea that I had to, and I ended up finding out a lot of really random interesting things through this rabbithole lol, including stuff about real life gerrymandering which I had no idea about
the text at 0:59 : "lmao imagine being a vice president of the United States, and what you're best known for is a corruprion term derived from your name. L bozo"
@@Deleted_Eevee two presidents of the us. They were very racist, Reagan missmanaged an AIDS epidemic and blamed ot on gay people and Nixon had that whole Watergate scandal
@@Deleted_Eevee Two presidents of the United States. Both republican. One was known for warmongering, and the other for an incident wherein he was caught spying on his political opposition. To balance the scales a bit, Clinton, a democrat, is best known for a sexual assault case in which several witnesses killed themselves. It's the origin of the suicide by two shots to the back of the head meme. Basically, there are far worse things your name can be tied to, and that goes for all parties in all countries.
@@Deleted_Eevee Former us presidents, i'm not too familiar with what Nixon messed up but Reagan sold a concept of trickle down economics (or Reagonomics) and used it to deregulate corporations at the expense of lower class citizens.
i wish simon tatham would add this to his online puzzle collection. it would be fun to play online on randomly generated boards that guaranteed to have a single solution
A couple years back my government teacher literally assigned us a video game that you have to gerrymander to win. It was funny but actually pretty educational.
you kinda skipped over explaining why that puzzle can’t have more or less than 5 regions - with 50 regions of 1 or 1 region of 50, it’ll just be majority rule - with 25 regions of 2, the top left cell has no purple neighbors and therefore cannot be in an untied region - with 2 regions of 25, there aren’t enough purple cells to achieve a majority in both regions - with 10 regions of 5, 6 regions of 3 purple are required, with the remainder being entirely yellow. however, it is impossible for the cell in the bottom left to be in a region with 2 other purple cells
good catch! he said all those numbers had solutions of x*y where y=2x this then expands to 2x^2 which is two times a square idk why it’s odd though, he said something about odd numbers being better?
sounds like a fun competitive game where you have a big board of randomised coloured tiles and players competing to create the most districts in their colour, the bigger the more points. Though you would probably have to add another dimension so that it isn't a pure puzzle solving game.
The part with 320 is a clear pattern, the lower the power of two the ratio is, the more common it is as a solution, search for long enough and I'm sure you can find x * y = 8x * y/8 and so on, you found it odd because you only saw two instances of 4 and chalked one up to be an edge case instead of the two being in a sequence
0:42 wikipedia - its intended pronunciation as named after Elbridge Gerry, is with a plosive [g] instead of the affricate [dʒ] which could appear so because of the front vowel [ɛ] following it
Not in this case. Gerrymandering is a separate problem, and won't be fixed by replacing FPTP with a better system. Gerrymandering _would_ be fixed by a switch to Proportional voting, but that's because switching to Proportional voting is specifically a change to the part of the political system that includes gerrymandering (it would replace the concept of "districts"). And it's quite possible to switch to Proportional voting without giving up FPTP; that's call "Party list PR".
Back in the 90s and 00s, the puzzle used to be called a "Kasian-style" puzzle, because they were related to the logic puzzle games created by Everett Kaser Software.
This reminds me of a layton riddle wher you had to win a game of, I thinks it's called the war in english, but you have worse cards than your opponent so you have to follow the same strategy of just winning with a small advantage when you win and loosing by a lot when you loose
1:00 quick text says “lmao imagine being a vice president of the United States, and what you’re best known for is a corruption term derived from your name. L bozo”
the tricolor puzzle at the end was fun, 3 sections with (3c 2m 2y), 2 sections with (0c 0m 7y), and 2 sections with (0c 4m 3y). I did manage to draw the arrangement too, but describing it in youtube comments doesn't seem efficient.
Another fun variant, you mentioned at the end the possibility of 3 distinct voter parties, but in the real world, there are also demographics that don't vote at all, and can be included without affecting the balance between the other parties
19:40 it sounds like 16 and 320 are forming their own sequence of numbers with their own switching factor of 4, which interleaves with the one you already found with switching factor of 2. At some point, you'll have to consider that you don't have an edge case but a perfectly formed separate sequence, especially since you didn't really explore that high (100's are tiny numbers by computational exploration standards) EDIT: nvm
It seems to me that a lot of the weird pattern shenanigans happen because of rounding up. I wonder if the pattern would be simpler if we said that for even numbers of districts and regions, getting a tie is equivalent to winning that region/board. Or in other words, if your goal was not to win, but to prevent the opponent from getting a strict majority.
somebody beat me to it -- i was working on a simplified grid-based gerrymandering puzzle game too, ive been inactive in developing it for a while but i started a few months ago
20:35 the other number in multiplication in a sequence 1, 5, 21... might be a sum of all powers of 4 up to certain power 1 = 4⁰ 5 = 4¹ + 4⁰ 21 = 4² + 4¹ + 4⁰ ... So if my conjecture is right, 87,040 might be the next "exception" number
87040 is also a 320-case: 85 * 2^10 = 340 * 2^8. More generally, (the sum from 0 to n of 4^n) * 4^(n+2) is the same as (4(the sum from 0 to n of 4^n)) * 4^(n+1)
@@tBagley43 Yes, that works, but it can be expressed with the explicit formula a(n) = 4^(n+1)*(sum from i=0 to (n-1) of 4^i), a(1) = 16, a(2) = 320, a(5) = 4^6 * (1+4+16+64+256) = 1396736.
@@rmrmarbleracing5372 oh yeah that's even easier, nice find. and actually you can express "sum from i = 0 to (n-1) of 4^i" more simply as "(4^n - 1)/3", so after distributing, the entire expression is: a(n) = (4^(2n+1) - 4^(n+1))/3. you could also reindex to make it a little nicer if you don't want to consider 0 as a trivial solution.
prime factorizations of the needed minority districts: 16 - 9=3^2 320 - 99=3^2*11 784 - 225=3^2*5^2 3536 - 945=3^3*5*7 5376 - 1419=3*11*43 10208 - 2655=3^2*5*59 13376 - 3465=3^2*5*7*11 16576 - 4275=3^2*5^2*19 20336 - 5229=3^2*7*83 36848 - 9405=3^2*5*11*19 44896 - 11439=3^2*31*41 48256 - 12285=3^3*5*7*13 95408 - 24165=3^3*5*179 113296 - 28665=3^2*5*7^2*13 122816 - 31059=3^2*7*17*29 143008 - 36135=3^2*5*11*73 191744 - 48375=3^2*5^3*43 225776 - 56925=3^2*5^2*11*23 267008 - 67275=3^2*5^2*13*23 270256 - 68085=3^2*5*17*89 296416 - 74655=3^3*5*7*79 343408 - 86445=3^2*5*17*113 393856 - 99099=3^2*7*11^2*13 398816 - 100359=3^5*7*59 466496 - 117315=3^3*5*11*79 525008 - 131985=3^2*5*7*419 630208 - 158355=3^4*5*17*23 697936 - 175329=3^2*7*11^2*23 793616 - 199305=3^2*5*43*103 864416 - 217035=3^2*5*7*13*53 869408 - 218295=3^4*5*7^2*11 921856 - 231435=3^2*5*37*139 948656 - 238149=3^2*47*563 one notable thing, all of them are divisible by 3, and all of them except 5376 - 1419 for some reason are divisible by 9 also all of them except 16 - 9, 320 - 99, 5376 - 1419, 20336 - 5229, 44896 - 11439, 122816 - 31059, 393856 - 99099, 398816 - 100359, 697936 - 175329, and 948656 - 238149 are divisible by 5 (and since 1419 is already an exception to this, 45 as well) notable patterns among exceptions of these rules: all the minority district amounts not divisible by 5 end in 9, all total district amounts where the minority district amount is not divisible by 5 end in 6 besides 320 (which is also the only total district amount to end in 0 in the whole list)
i made a reply with all of these numbers in both hex and binary but then i lost internet connection just before i posted and it got deleted i'm not doing all that again so here's the only notable thing i found: all the total district numbers are divisible by 16, which makes sense since they all have (a,b)-(4a,b/4) pairs so they must be divisible by 4*4=16
14:20 “7*20” appears, which, while I know is 140, looks like “720” (that I almost ALWAYS pronounce as “Seven-Twenty”, for short), which is my favorite number (as you might be able to tell from my UA-cam channel name and UA-cam handle name)!!!!
17:07 (You also happened to begin the count of “Prime number times 4” with the multiple of 59, which is *another* one of my unlucky numbers. Also, I'm sorry that I comment so much. However, there will probably be even MORE comments from me, because I can't control myself!)
The "4-apart" pattern you've basically explained yourself already - it's the numbers that are divisible by 2 but not divisible by 4, following the formula 4n+2. The pattern breaks on double squares, because then two solutions given by the formula coincide. As for 320 there's two effects here: one that prefers oddsover evens (because adding half a square in a dimension is cheaper than adding a full square) at odds with "factoring a numbers such that the sum of factors is minimal", which favors two factors that are close to each other. The second effect starts to dominate for large numbers with only small odd factors, such as 320 = 5*64, 5
17:45 I tried throwing those numbers into OEIS and it found just one sequence that goes 1,2,18,50,98,162,242,338,450,... Could you verify if your system also finds those additional numbers?
I was going to comment this if no one else did. I went right to the OEIS when he started talking about "random" sequences of numbers. Always worth checking that. Not sure how "Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals." might connect to these puzzles but it could be worth looking at.
Are there any 8x * x/8 solutions? And maybe the 16 case can be generalized to show that the pattern holds for other powers of two, but frequently generates multiple non-unique solutions.
During the section about the weird double minimas at 20:26, I noticed that each of the odd factors listed are 4x + 1 of the previous(or the next power of 4 + the previous number). so I did some testing and these were the results: number being tested factor, factor, smallest result of test (factors are only listed when they lead to the smallest result in testing) 16 1, 16, 36 4, 4, 36 320 5, 64, 396 16, 20, 396 5376 21, 256, 5676 64, 84, 5676 87040 272, 320, 88228 1396736 341, 4096, 1401516 1024, 1364, 1401516 22364160 4480, 4992, 22383108 357892096 5461, 65536, 357968556 16384, 21844, 357968556 5726535680 69904, 81920, 5726839332 91625619456 299008, 306432, 91626830340 1466014105600 1154560, 1269760, 1466018954244 23456242466816 1398101, 16777216, 23456262040236 4194304, 5592404, 23456262040236 375299946577920 19249152, 19496960, 375300024070148 6004799413682176 22369621, 268435456, 6004799726856876 67108864, 89478484, 6004799726856876 96076791692656640 304087040, 315951616, 96076792932733956 1537228671377473536 1228333056, 1251475456, 1537228676337090564 After this I was getting integer overflow errors. Its also worth noting that 87040 second smallest outcome is a double minima like the others. If i had to guess why this sort of works then I'd probably say that since it works for 16 if you were to time both factors by 4 but keep the odd one odd you might expect it to also work. Hope this helps.
Looks like factors of the form 2k+2 have a strong tendency to have solutions that are k better than the norm. Does this continue for larger k, and for larger second factors, as well?
Gerrymandering was invented by Elbridge Gerry in 1812. That sounds like the Thomas Running joke, but it's actually real this time
Shrapnel was invented by John General Shrapnel
@@cubee4108 The German chocolate cake was invented by an English-American chocolate maker named Samuel German
Mewing was invented by doctor John mew
@@cubee4108 Henry but actually close. I can't tell if this is a joke or not sorry
@@m4rcyonstation93Henry but actually close was invented by close but actually henry
Oh wow, I really like the idea of turning gerrymandering into a puzzle! It reminds of puzzles like Nonograms, KenKen, and so on. And, it can teach you a thing or two about the real world's districting-patterns!
UA-cam legend located 🫵(I’m a fan)
🤜🤛
cary keeprigging houseofrepresentativeselections
It reminds me of games like minesweeper.
yeah, I love those types of puzzles! especially nonograms, those sorts of things are what especially inspired me to make this into a type of puzzle! it was just such a funny idea that I had to, and I ended up finding out a lot of really random interesting things through this rabbithole lol, including stuff about real life gerrymandering which I had no idea about
3:31 I don't want to tie this colors to political parties *Proceeds to use the colors of the political parties of my country*
To american political party***
It's also the colours of UKIP
@@howtonamevar Reform and Libertarian are right there, sure they're third parties but they're still parties
I'm sorry, do you live in Warioland?
What’s the country op
3:35 now it’s Wario vs Waluigi.
Which is, to be fair, more important topic than any politics anywhere in the world.
I’m glad all Waluigi can win every single time, even with Wario’s popularity.
the text at 0:59 : "lmao imagine being a vice president of the United States, and what you're best known for is a corruprion term derived from your name. L bozo"
I mean there are worse things to be remembered for. Like Reagan... Or Nixon... The list goes on among politicians of basically every country.
@@magmati55could you by any chance tell me who those people are I have no clue
@@Deleted_Eevee two presidents of the us. They were very racist, Reagan missmanaged an AIDS epidemic and blamed ot on gay people and Nixon had that whole Watergate scandal
@@Deleted_Eevee Two presidents of the United States. Both republican. One was known for warmongering, and the other for an incident wherein he was caught spying on his political opposition.
To balance the scales a bit, Clinton, a democrat, is best known for a sexual assault case in which several witnesses killed themselves. It's the origin of the suicide by two shots to the back of the head meme.
Basically, there are far worse things your name can be tied to, and that goes for all parties in all countries.
@@Deleted_Eevee Former us presidents, i'm not too familiar with what Nixon messed up but Reagan sold a concept of trickle down economics (or Reagonomics) and used it to deregulate corporations at the expense of lower class citizens.
The New York Times made a gerrymandering puzzle like this on their website a while ago, funnily enough, also with yellow and purple colors.
I actually love Gerrymandering puzzles. I’ve played two flash games where that’s the mechanic, and it’s really fun
there was a flash game called the redistricting game that I miss dearly
It might be on Flashpoint, have you checked?
I remember a mobile game with the same idea as this video too. The names of the levels were wild. If only it was still around
Its on flashpoint so you can still play it with that, I just checked
i wish simon tatham would add this to his online puzzle collection. it would be fun to play online on randomly generated boards that guaranteed to have a single solution
4:13 Invalid solution! There's two V and T pentominoes- wait, this isn't pentomino pathfinding... anyway
Haha! Pentamino puzzles are awesome
the grid you made for the minimum cells to win looks like how minecraft tnt works when its explosion size gets really big.
IS UR PFP BY SINCLAIRFAN1 I MISS HER ART
A couple years back my government teacher literally assigned us a video game that you have to gerrymander to win. It was funny but actually pretty educational.
I'd love to see Numberphile examine these weird cases. It sounds interesting
you kinda skipped over explaining why that puzzle can’t have more or less than 5 regions
- with 50 regions of 1 or 1 region of 50, it’ll just be majority rule
- with 25 regions of 2, the top left cell has no purple neighbors and therefore cannot be in an untied region
- with 2 regions of 25, there aren’t enough purple cells to achieve a majority in both regions
- with 10 regions of 5, 6 regions of 3 purple are required, with the remainder being entirely yellow. however, it is impossible for the cell in the bottom left to be in a region with 2 other purple cells
This went from funny puzzle game about politics to mathematics real quick
This video is NOT about the politics and nuances of REAL LIFE gerrymandering!
Took the words right out of my mouth (ew)
What title did it show you? The title I saw ("Let's turn America's broken election system into a puzzle genre!") seems honest to me.
@@Tzizenorecyou're kind of foolish
18:19, those numbers are all two times an odd perfect square. I’m not sure if that’s significant or not but I’d guess that it is.
good catch!
he said all those numbers had solutions of x*y where y=2x
this then expands to 2x^2 which is two times a square
idk why it’s odd though, he said something about odd numbers being better?
wait am I crazy but isn't it just two times a perfect square?
32 = 16*2 = 4²*2
50 = 25*2 = 5²*2
72 = 36*2 = 6²*2
@@colecube8251 the solutions are only with odd perfect squares
i’m not sure if that’s what op was talking about thouh
sounds like a fun competitive game where you have a big board of randomised coloured tiles and players competing to create the most districts in their colour, the bigger the more points. Though you would probably have to add another dimension so that it isn't a pure puzzle solving game.
The part with 320 is a clear pattern, the lower the power of two the ratio is, the more common it is as a solution, search for long enough and I'm sure you can find x * y = 8x * y/8 and so on, you found it odd because you only saw two instances of 4 and chalked one up to be an edge case instead of the two being in a sequence
0:42 wikipedia - its intended pronunciation as named after Elbridge Gerry, is with a plosive [g] instead of the affricate [dʒ] which could appear so because of the front vowel [ɛ] following it
It's not called garymandering
I believe everest is a similar situation
@@BookWyrmOnAStringyea George everest pronounced his name /ivrɪst/ but the modern pronunciation is (depending on your dialect) /ɛvrɪst/
no, it’s gerrymandering because it’s a type of jerry-rigging (this is a joke)
Thanks for this! I have a family member named Gerry and find myself having to explain that a lot.
John Gerrymander really cooked with this
Actually it was Elbridge Gerry
@Nerdy1729 Sureeeeeeeeeeeeeeeee
Actually it was Jerry M. Anders
@@anthonybutori4149 ah shit mb mb
Gohn Gerry Ginglemander Smith
As a person who plays around in Excel (well, actually LibreOffice Calc but whatever) for fun, this video is chewy and full of flavor.
33 seconds and we're into the content.
Bless you weird YT Games/Maths person, quality content like yours is a rarity.
that ending puzzle was so well designed, finding the logic needed to progress each time with the 2x2 rule was so interesting but completely logical
Yeah, all of these problems go back to First Past the Post… (CGP Grey)
Sadly, he's stuck on transferrable vote and applies that bias to his later videos.
@@tristanridley1601 As opposed to what?
Not in this case. Gerrymandering is a separate problem, and won't be fixed by replacing FPTP with a better system.
Gerrymandering _would_ be fixed by a switch to Proportional voting, but that's because switching to Proportional voting is specifically a change to the part of the political system that includes gerrymandering (it would replace the concept of "districts"). And it's quite possible to switch to Proportional voting without giving up FPTP; that's call "Party list PR".
Back in the 90s and 00s, the puzzle used to be called a "Kasian-style" puzzle, because they were related to the logic puzzle games created by Everett Kaser Software.
I really wanted to see the minimum percentages you need to win together with the total amount you need
This reminds me of a layton riddle wher you had to win a game of, I thinks it's called the war in english, but you have worse cards than your opponent so you have to follow the same strategy of just winning with a small advantage when you win and loosing by a lot when you loose
I did this as an assignment in school once lol. We were given a map and were told to gerrymander it. It was fun!
1:00 quick text says “lmao imagine being a vice president of the United States, and what you’re best known for is a corruption term derived from your name. L bozo”
the tricolor puzzle at the end was fun, 3 sections with (3c 2m 2y), 2 sections with (0c 0m 7y), and 2 sections with (0c 4m 3y). I did manage to draw the arrangement too, but describing it in youtube comments doesn't seem efficient.
The Electoral Circus: The Game
This feels a lot like sudoku shading puzzles, I like it
Someone needs to turn this into a video game, where the boards are randomly generated
Another fun variant, you mentioned at the end the possibility of 3 distinct voter parties, but in the real world, there are also demographics that don't vote at all, and can be included without affecting the balance between the other parties
Don't let politicians play this game
19:40 it sounds like 16 and 320 are forming their own sequence of numbers with their own switching factor of 4, which interleaves with the one you already found with switching factor of 2. At some point, you'll have to consider that you don't have an edge case but a perfectly formed separate sequence, especially since you didn't really explore that high (100's are tiny numbers by computational exploration standards)
EDIT: nvm
It seems to me that a lot of the weird pattern shenanigans happen because of rounding up. I wonder if the pattern would be simpler if we said that for even numbers of districts and regions, getting a tie is equivalent to winning that region/board. Or in other words, if your goal was not to win, but to prevent the opponent from getting a strict majority.
this is some cracking the cryptic material
this is what i thought too, the second the cells started getting highlighted different colors lol
somebody beat me to it --
i was working on a simplified grid-based gerrymandering puzzle game too, ive been inactive in developing it for a while but i started a few months ago
Will it be a computer game? I think this concept sounds pretty fun and idk if the guy who made the video will actually make one
I essentially made this game for my computer science final project in high school. Love messing around with gerrymandering puzzles.
Absolutely wonderful video! Would you consider making it into an actual mini videogame?
now I really want to play a gerrymandering puzzle game
20:35 the other number in multiplication in a sequence 1, 5, 21... might be a sum of all powers of 4 up to certain power
1 = 4⁰
5 = 4¹ + 4⁰
21 = 4² + 4¹ + 4⁰
...
So if my conjecture is right, 87,040 might be the next "exception" number
According to another comment, yes, that is the next exception number!
21:32
Yeah, well what if instead of a bean stalk, we had a bean trellis?
My high school government teacher did this. It’s pretty fun!
Great video, thanks for sharing -- any tips on constructing gerrymander grids with a unique solution?
please turn this into an actual game!
87040 is also a 320-case: 85 * 2^10 = 340 * 2^8.
More generally, (the sum from 0 to n of 4^n) * 4^(n+2) is the same as (4(the sum from 0 to n of 4^n)) * 4^(n+1)
it looks like a recurrence relation:
a(0) = 0
a(1) = 16
a(n) = 20*a(n-1)-64*a(n-2)
so the next number ought to be 1396736, could you check that?
@@tBagley43 Yes, that works, but it can be expressed with the explicit formula a(n) = 4^(n+1)*(sum from i=0 to (n-1) of 4^i), a(1) = 16, a(2) = 320, a(5) = 4^6 * (1+4+16+64+256) = 1396736.
@@rmrmarbleracing5372 oh yeah that's even easier, nice find. and actually you can express "sum from i = 0 to (n-1) of 4^i" more simply as "(4^n - 1)/3", so after distributing, the entire expression is: a(n) = (4^(2n+1) - 4^(n+1))/3. you could also reindex to make it a little nicer if you don't want to consider 0 as a trivial solution.
list of all non-sequence numbers below 1 million:
16 - (1, 16), (4, 4) - 9 districts
320 - (5, 64), (16, 20) - 99 districts
784 - (16, 49), (28, 28) - 225 districts
3536 - (17, 208), (52, 68) - 945 districts
5376 - (21, 256), (64, 84) - 1419 districts
10208 - (29, 352), (88, 116) - 2655 districts
13376 - (64, 209), (88, 152) - 3465 districts
16576 - (37, 448), (112, 148) - 4275 districts
20336 - (41, 496), (124, 164) - 5229 districts
36848 - (112, 329), (188, 196) - 9405 districts
44896 - (61, 736), (184, 244) - 11439 districts
48256 - (128, 377), (208, 232) - 12285 districts
95408 - (89, 1072), (268, 356) - 24165 districts
113296 - (97, 1168), (292, 388) - 28665 districts
122816 - (101, 1216), (304, 404) - 31059 districts
143008 - (109, 1312), (328, 436) - 36135 districts
191744 - (256, 749), (428, 448) - 48375 districts
225776 - (137, 1648), (412, 548) - 56925 districts
267008 - (149, 1792), (448, 596) - 67275 districts
270256 - (304, 889), (508, 532) - 68085 districts
296416 - (157, 1888), (472, 628) - 74655 districts
343408 - (169, 2032), (508, 676) - 86445 districts
393856 - (181, 2176), (544, 724) - 99099 districts
398816 - (352, 1133), (484, 824) - 100359 districts
466496 - (197, 2368), (592, 788) - 117315 districts
525008 - (209, 2512), (628, 836) - 131985 districts
630208 - (229, 2752), (688, 916) - 158355 districts
697936 - (241, 2896), (724, 964) - 175329 districts
793616 - (257, 3088), (772, 1028) - 199305 districts
864416 - (544, 1589), (908, 952) - 217035 districts
869408 - (269, 3232), (808, 1076) - 218295 districts
921856 - (277, 3328), (832, 1108) - 231435 districts
948656 - (281, 3376), (844, 1124) - 238149 districts
784 is even weirder than 320 because (16,49) and (28,28) aren't even an (a,b)-(4a,b/4) pair
the numbers that don't form pairs like this are 784, 13376, 36848, 48256, 191744, 270256, 398816, 864416, 1082848, 1565120, 1581664, 2020928, 2762560, 2766368, 3060736, and 3981712
(28,28) does form an (a,b)-(4a,b/4) pair with (7,112), but that requires 228 minority districts, while (28,28) only requires 225 minority districts
What if we write these numbers in binary
prime factorizations of the needed minority districts:
16 - 9=3^2
320 - 99=3^2*11
784 - 225=3^2*5^2
3536 - 945=3^3*5*7
5376 - 1419=3*11*43
10208 - 2655=3^2*5*59
13376 - 3465=3^2*5*7*11
16576 - 4275=3^2*5^2*19
20336 - 5229=3^2*7*83
36848 - 9405=3^2*5*11*19
44896 - 11439=3^2*31*41
48256 - 12285=3^3*5*7*13
95408 - 24165=3^3*5*179
113296 - 28665=3^2*5*7^2*13
122816 - 31059=3^2*7*17*29
143008 - 36135=3^2*5*11*73
191744 - 48375=3^2*5^3*43
225776 - 56925=3^2*5^2*11*23
267008 - 67275=3^2*5^2*13*23
270256 - 68085=3^2*5*17*89
296416 - 74655=3^3*5*7*79
343408 - 86445=3^2*5*17*113
393856 - 99099=3^2*7*11^2*13
398816 - 100359=3^5*7*59
466496 - 117315=3^3*5*11*79
525008 - 131985=3^2*5*7*419
630208 - 158355=3^4*5*17*23
697936 - 175329=3^2*7*11^2*23
793616 - 199305=3^2*5*43*103
864416 - 217035=3^2*5*7*13*53
869408 - 218295=3^4*5*7^2*11
921856 - 231435=3^2*5*37*139
948656 - 238149=3^2*47*563
one notable thing, all of them are divisible by 3, and all of them except 5376 - 1419 for some reason are divisible by 9
also all of them except 16 - 9, 320 - 99, 5376 - 1419, 20336 - 5229, 44896 - 11439, 122816 - 31059, 393856 - 99099, 398816 - 100359, 697936 - 175329, and 948656 - 238149 are divisible by 5 (and since 1419 is already an exception to this, 45 as well)
notable patterns among exceptions of these rules: all the minority district amounts not divisible by 5 end in 9, all total district amounts where the minority district amount is not divisible by 5 end in 6 besides 320 (which is also the only total district amount to end in 0 in the whole list)
i made a reply with all of these numbers in both hex and binary but then i lost internet connection just before i posted and it got deleted
i'm not doing all that again so here's the only notable thing i found:
all the total district numbers are divisible by 16, which makes sense since they all have (a,b)-(4a,b/4) pairs so they must be divisible by 4*4=16
May I know what is the districts required for 87040?
BTW for the chapter feature to work I believe you need to have 0:00 also as a chapter
12:08 I didn't even NOTICE that, but you're right!!!! 😂😂🤣🤣
14:20 “7*20” appears, which, while I know is 140, looks like “720” (that I almost ALWAYS pronounce as “Seven-Twenty”, for short), which is my favorite number (as you might be able to tell from my UA-cam channel name and UA-cam handle name)!!!!
16:54 And one of my unlucky numbers, 338, ALSO appears, ironically enough for my next reply after that...
17:07 (You also happened to begin the count of “Prime number times 4” with the multiple of 59, which is *another* one of my unlucky numbers. Also, I'm sorry that I comment so much. However, there will probably be even MORE comments from me, because I can't control myself!)
18:36 More “7*20”!!!!
You should make a sudoku variant with this
Please make this a full research project I would love to see this generated near infinite time and conclusions be developed for this
17:52 don't forget about 320
ahh, I watched more of the video, and you mentionned it.
12:39, purple isn't the minority in either of these
0:58 Elbridge Gerry was actually pronounced with a g, but due to the word gerrymander spreading people started mispronouncing it as a j.
6:14 this actually looks like the exact minimum number of purples!
The "4-apart" pattern you've basically explained yourself already - it's the numbers that are divisible by 2 but not divisible by 4, following the formula 4n+2. The pattern breaks on double squares, because then two solutions given by the formula coincide. As for 320 there's two effects here: one that prefers oddsover evens (because adding half a square in a dimension is cheaper than adding a full square) at odds with "factoring a numbers such that the sum of factors is minimal", which favors two factors that are close to each other. The second effect starts to dominate for large numbers with only small odd factors, such as 320 = 5*64, 5
18 50 98 162 and 242 are all an odd square times two (2 times 9 25 49 81 and 121 respectively)
I never knew gerrymandering was a portmanteau
In high school my government teacher had us play a little gerrymandering game on some website to learn about it.
17:45 I tried throwing those numbers into OEIS and it found just one sequence that goes 1,2,18,50,98,162,242,338,450,...
Could you verify if your system also finds those additional numbers?
that is 2*each odd perfect square except the 1 fsr (2 * 1², 2 * 3², 2 * 5², 2 * 7², …)
I was going to comment this if no one else did. I went right to the OEIS when he started talking about "random" sequences of numbers. Always worth checking that. Not sure how "Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals." might connect to these puzzles but it could be worth looking at.
1:06 In this house, we prefer our fusions fancy god earrings style. Not some rocks.
There’s this game on cool math games that’s kind of like this called districts, it is fun af
17:45 All of these numbers have squarefree part 2.
... yet so does 338, which you showed on the highest sloped line at 16:55.
Are there any 8x * x/8 solutions? And maybe the 16 case can be generalized to show that the pattern holds for other powers of two, but frequently generates multiple non-unique solutions.
3:22 i think the labels are wrong?
It's not broken. It works as intended. And it works really well.
If you ever make an app that puts this into a game please let me know im ready to absoltuely eat ny day up woth these puzzles
6:08 why is this 3 red? Are you plotting something in particular?
I legit did this for my highschool US history final 2 years ago 💀💀💀
I found this pretty interesting even though I barely understand it
WHY IS YOUR VOICE SO SIMILAR TO CARY KH
How do we know the amount of districts we can use?
this would make a really fun mobile puzzle game
someone show this to matt parker and numberphile, this is very cool
any chance you release this puzzle as an app or a website?
Making this game looks fun, you could do it with graph theory, and a breath first search
1 *4+1 = 5, 5 *4+1 = 21, and the multiplicand is 4^n, or OEIS A002450
Or you could use Proportional Representation.
Loved how the last puzzle can be solved for blue it was hard but I did it
13:42 should be 3 x 21 right?
gerrymeowering
There is a fun app that does this type of puzzle called Gerrymander
1:08 : you mean dragon ball Z fused, right?
Cool video!
hasnt this already been done before? i remember playing a mobile game that was really good based on it
During the section about the weird double minimas at 20:26, I noticed that each of the odd factors listed are 4x + 1 of the previous(or the next power of 4 + the previous number). so I did some testing and these were the results:
number being tested
factor, factor, smallest result of test (factors are only listed when they lead to the smallest result in testing)
16
1, 16, 36
4, 4, 36
320
5, 64, 396
16, 20, 396
5376
21, 256, 5676
64, 84, 5676
87040
272, 320, 88228
1396736
341, 4096, 1401516
1024, 1364, 1401516
22364160
4480, 4992, 22383108
357892096
5461, 65536, 357968556
16384, 21844, 357968556
5726535680
69904, 81920, 5726839332
91625619456
299008, 306432, 91626830340
1466014105600
1154560, 1269760, 1466018954244
23456242466816
1398101, 16777216, 23456262040236
4194304, 5592404, 23456262040236
375299946577920
19249152, 19496960, 375300024070148
6004799413682176
22369621, 268435456, 6004799726856876
67108864, 89478484, 6004799726856876
96076791692656640
304087040, 315951616, 96076792932733956
1537228671377473536
1228333056, 1251475456, 1537228676337090564
After this I was getting integer overflow errors. Its also worth noting that 87040 second smallest outcome is a double minima like the others. If i had to guess why this sort of works then I'd probably say that since it works for 16 if you were to time both factors by 4 but keep the odd one odd you might expect it to also work. Hope this helps.
20:33 shouldn’t the bottom right power of two be 2^6?
So (floor(amountofgroups/2)+1)(floor(sizeofgroups/2)+1) is the minimum amount?
If you thought the Electoral College was bad, just wait until you see this!
Those are such weird and interesting patterns. This needs more research, for sure
3:31 ah yes, the Warios VS the Waluigis
maybe not important but Gerry's name is pronounced with a hard G, like Gary
Looks like factors of the form 2k+2 have a strong tendency to have solutions that are k better than the norm. Does this continue for larger k, and for larger second factors, as well?
Do the 3-colored ones follow majority-rule or first past the post?
That puzzle was fun!
I always play this in math clasa
gerrymander? i hardly know her