Yeah! Moshe and Aloysius especially shouldn't be that hard. Aloysius is one of the many longer forms of Al (Allen, Alex, Albert, Algernon, Alfred, etc.)
Matt mispronouncing Łukasz makes me think of a bit of maths/computer science history. In 1924, Jan Łukasiewicz invented prefix notation for operations. For example, you would write "2 + 3" as "+ 2 3", and "4 + (5 - 2)" would be "+ 4 - 5 2" - you see, if you know how many things to with each operator, this notation is completely unambiguous. Now, where many inventions get named after their inventor (see "Turing machines" after Alan Turing, or "Dijkstra's algorithm" after Edsger Dijksta), with Łukasiewicz... not so much. Instead, this notation is most commonly referred to as "Polish notation", as people couldn't be bothered with learning how to pronounce "Łukasiewicz".
Hey Matt, thanks for mentioning my spreadsheet, I love that you call it an "honorable mention" :) But I think, you forgot to link to the spreadsheet in the description 😬 Kind regards, Simon
A 8x8 grid contains 7x7 grids, but those do not necessarily contain 25 red and 24 green tokens. So, you have to show that there are no square-free solutions for any combination of red and green tokens. (Luckily, there are none)
An 8x8 has 15 fields more than a 7x7. Even if they are all red there are at least 17 red fields in the 7x7 subgrid for the 8x8 to be balanced. So you only need to check if there are no solutions in a 7x7 for the number of tokens of one color between 17 and 24 in order to proof that there is no balanced 8x8 solution. However this is not a proof that there is no "balanced" 9x9 solution.
I feel like this series is genuinely really enjoyable for Matt cause it's like having a classroom of 2000+ attentive students and no actual curriculum to stick to besides "do some fun maths things"
11:19 - That's not a solution there is a square, pieces 2, 4, 12 and 14 are all blank and form a medium 90. Also 8, 15, 17, 24 form a small right windmill with pieces.
9:21 This is not correct with the original puzzle: There you demanded (by the nature of a two player game) the same number of red and green tokens (give or take one for odd sizes). That way, a balanced 8x8 need not have a balanced 7x7 in it! However, it is in fact true that there does not even exist a 7x7 solution if we ignore balance (at least two fields will need to be left empty). This then indeed shows that no larger square (or rectangle) can be solved woth or without balancedness condition.
I was wondering that myself. For example in the 4x4 case, you can have: xxxx oooo oxxo ooxx In this case, the bottom left 3x3 isn't balanced (though the other 3 are). I could imagine a case where there is a valid n*n containing no balanced (n-1)*(n-1) grids.
That's an interesting question. If the grid is balanced, can you always find a balanced subgrid of any size. Maybe some type of argument with mean value theorem would work, although this is discrete not continuous
If there is no balanced solution (or "as near to balanced as possible" for odd n), there cannot be an unbalanced solution either, because you have to add more tiles of a color that already has some squares.
@@screwaccountnames I believed you for a second, but I'm not sure I'm convinced. Suppose (wlog) there is a balanced 7x7 grid with 24 white tiles and 25 black tiles. Suppose that this grid contains a single square of 4 white tiles. One could substitute a single white tile for a black tile and potentially remove that square, leading to a 7x7 grid with 23 white and 26 black tiles, i.e. a valid but unbalanced grid.
I made my life easy by using a randomized algorithm which simply generates random placements until one without squares is found. Worked very well for 5x5 at least.
I wrote a program (in Haskell) that would've found all of them... it was just really, really slow. I wanted to try and optimize it a bit first, then send it in, but I forgot. It first calculated all of the potential squares for that size, then generated every possible game (with a balanced number of stones), then checked all of the games against those potential squares, filtered out those that had none, and printed them out. It spit out a size 5 solution every few seconds (after you gave it a while to get going), but it never got a size 6 solution. I gave it like 30 minutes.
So, I entered this one, and did it free-hand right on the submission, didn't write anything down, just winged it, thought I had checked my answer thoroughly, and would be interested in seeing what I put in, after this solutions video came out so I could see where I went wrong, as it has been a little while... but I can't as far as I can tell.
The 24 solutions for 6x6 looks suspiciously like the 18v18 case with no duplicate-cancellation, ignoring the 16 solutions for 17v19. In which case, the 0 solutions for the 7x7 board isn't conclusive for the 8x8 board - there being no 24v25 solutions doesn't immediately guarantee the absence of 23v26 solutions (it may be possible to prove that any solution with unequal numbers of the two colours can be turned into another solution by flipping a token of the majority colour, but it's not obvious that it's even true, let alone provable). So it would need to be shown in some way that you couldn't get a balanced 8x8 solution (32v32) by overlapping unbalanced (23v26 or more extreme) 7x7 solutions. Presumably at some point the situation arises where you can't fit enough of one colour onto an NxN board even before considering the second colour - if you could prove that you can't fit 25 counters on a 7x7 in a square-free fashion, that would immediately prove that 7x7 and all larger boards are impossible.
The puzzle was to fill a 5x5 grid and form no squares. A solution confirms that cooperating players can engineer a tie game. But can a tie come about in normal play? In other words, does neither player have a winning strategy?
Steve the Cat Couch If my computer analysis is correct, the player who goes second can force a win; i.e., they can force the first player to eventually make a square. First player’s best hope is to make his first move to a corner. If second player does not also play to a corner, first player can then force a tie.
The "windmills" (2:27) made me think I might not have submitted a valid answer - I had checked for squares where the corners are "knight move" away from each other (the small windmills), but I don't remember checking for the large windmills. Fortunately, according to the Java widget and spreadsheet, my submission WAS a valid answer.
He means reflections and rotations. A valid solution could be identical to another except they are rotated 90 degrees. Many people would prefer to remove those "duplicates" as they don't bring a new unique answer, just the same logic turned on its head.
@@rilesmcgiles1145 I see. He said "there *are* fewer than 24" which I thought implied there's no correct way to count 24, and they must've counted the same square twice.
@@aaronthompson2827 I got 24 as well including both symmetries and swapping the counter colours round. Matt might have been counting solutions that just exchange counter colours as duplicates. Similarly, the 2704 for 5x5 includes symmetric but not solutions that have 12 green counters as opposed to 13 green (or vice versa) - including both give 5408 for 5x5
@@rilesmcgiles1145 If a solution could be rotated by 90 degrees, it would have to have a lot of squares. Of course, reflections and 180-degree rotations are still valid.
Nope, and that's the only one I've gotten wrong so far, so I really want to see what the right answer is. (But I misplaced my answer, so I guess I won't really be able to see how close I was. Maybe I'll try again.)
No, an 8 by 8 does not necessarily contain a 7 by 7. For a 7 by 7 grid, there are 49 cells and each player must thus make 24 or 25 moves. An 8 by 8 grid has 64 squares and thus each player must make 32 moves, however 15 of these can be by one player along the 2 edges. In fact, they could place 28 squares all around the edge. Thus any corner of the grid to make a 7 by 7 grid will have only 17 of their squares while it has 32 of the others players squares. Sure, this wont produce a valid solution, but that doesn't mean that it is a trivial case of an 8 by 8 must have a 7 by 7 in it.
Now, how many solutions are there for a 7x7 BUT, there are 3 colors of tokens now. Now how big can you make the grid. and what if there are 4 players, or 5. If we take the sequence of the largest n×n square possible for 1, 2, 3, ... amount of tokens: {n_1, n_2, n_3 ...} what is that sequence? (we know it starts with i=1, n=1. then i=2, n=6, {1, 6, ...} but then?)
Idk why but i just completely forgot to check for the Small Windmills entirely. I almost got them all covered up by accident too even. It's a shame too, because I remember having "extra" tokens left and thinking, "aha, the trick is that you HAVE to place these extras and they might be forced to make a square. But sadly, no..
I noticed what he calls the windmills while solving, but somehow I calculated them to be rhombuses and my submission included them 😒😒 Btw, what is the formula for weighted points ? I got 2000 on puzzle 9 and 500 (I think) on this, and my weighted score is 2100. Apparently the weight of the most recent is 100% and the immediate previous puzzle is 80%, but I don't have enough data to confirm
But they didn't call them hipsters, there were hippies, but then I'm not sure it was the hippies that used the word 'square' in this manner exclusively, there were also the beatniks, and other subcultures
Wow! 5Xinfinity nice... I like the idea of toy LEGO Danish nationalism or whatever hahaha... anyway playing the game against someone with some weird set of rules on the 5Xinfinite board would be interesting perhaps to pursue with game theory etc
LOL, I got 500 points and thought that meant I solved the puzzle. I don't even know what I did wrong. I think I didn't think about squares at angles, that screwed me over.
@@leefisher6366 Follow the "All details about Matt Parker's Maths Puzzles are on the Think Maths website" link in the description. Click on the "League Table And Scoring System" link. In the "Search by name..." field in the table, type the name you used in the submission form for the puzzle. The points are in the given week column, but only the past 6 weeks contribute to your total, so earlier weeks are not displayed.
Gah! I wish I could see what it was I missed. I was sure I got all of them. I even went through and looked for every square from each coin that could make that square.
Makes sense, since he can't really know their gender, but he also does it with his assistants, which is strange (did his assistants all ask for they pronouns? Unlikely)
It's honestly nice, in situations where the gender of someone is useless information (most times), especially when the person is someone you don't personally know, I much prefer they pronouns. It's less presumptuous
Hey, he's a mathematician. If he can choose to use an abstraction that's always correct, instead of a guess that looks correct but has hidden exceptions, you know he's going to go for it.
I love how everytime you pronounce 'solutions', you say it in a tone that makes me think you're not quite sure if it really is solutions.
It's unfortunate, but most Australians speak with an upwards inflection. In this case though it's completely deliberate and I love it every time.
They are "Parker 'solutions'"
Łukasz = WOO-käsh
Moshe = MO-sheh
Xu = Shoo
Aloysius = æl-oh-ISH-us
Something something Parker Pronunciations
Yeah! Moshe and Aloysius especially shouldn't be that hard. Aloysius is one of the many longer forms of Al (Allen, Alex, Albert, Algernon, Alfred, etc.)
Matt mispronouncing Łukasz makes me think of a bit of maths/computer science history. In 1924, Jan Łukasiewicz invented prefix notation for operations. For example, you would write "2 + 3" as "+ 2 3", and "4 + (5 - 2)" would be "+ 4 - 5 2" - you see, if you know how many things to with each operator, this notation is completely unambiguous.
Now, where many inventions get named after their inventor (see "Turing machines" after Alan Turing, or "Dijkstra's algorithm" after Edsger Dijksta), with Łukasiewicz... not so much. Instead, this notation is most commonly referred to as "Polish notation", as people couldn't be bothered with learning how to pronounce "Łukasiewicz".
@@PhilBagels To be fair, I'd heard the name Aloysius before I saw it written out, and I was really surprised to see it was spelled that way.
@@daanwilmer You watched that Computerphile video too, didn't you?
@@daanwilmer Well that solves a 4 year mystery of why it was called polish notation in my comp sci class
Hey Matt, thanks for mentioning my spreadsheet, I love that you call it an "honorable mention" :) But I think, you forgot to link to the spreadsheet in the description 😬 Kind regards, Simon
A 8x8 grid contains 7x7 grids, but those do not necessarily contain 25 red and 24 green tokens. So, you have to show that there are no square-free solutions for any combination of red and green tokens. (Luckily, there are none)
I was going to say the same thing!
Excellent point!
An 8x8 has 15 fields more than a 7x7. Even if they are all red there are at least 17 red fields in the 7x7 subgrid for the 8x8 to be balanced.
So you only need to check if there are no solutions in a 7x7 for the number of tokens of one color between 17 and 24 in order to proof that there is no balanced 8x8 solution.
However this is not a proof that there is no "balanced" 9x9 solution.
"Oh, it appears in the little window with me!"
This part had me laughing a lot ^^
I feel like this series is genuinely really enjoyable for Matt cause it's like having a classroom of 2000+ attentive students and no actual curriculum to stick to besides "do some fun maths things"
11:19 - That's not a solution there is a square, pieces 2, 4, 12 and 14 are all blank and form a medium 90. Also 8, 15, 17, 24 form a small right windmill with pieces.
I noticed that too. No bonus points for using LEGO when your solution isn't even correct!
And 3, 10, 12, 19 are blank as well (1 row above the small right windmill with pieces)
9:21 This is not correct with the original puzzle: There you demanded (by the nature of a two player game) the same number of red and green tokens (give or take one for odd sizes). That way, a balanced 8x8 need not have a balanced 7x7 in it!
However, it is in fact true that there does not even exist a 7x7 solution if we ignore balance (at least two fields will need to be left empty). This then indeed shows that no larger square (or rectangle) can be solved woth or without balancedness condition.
I was wondering that myself. For example in the 4x4 case, you can have:
xxxx
oooo
oxxo
ooxx
In this case, the bottom left 3x3 isn't balanced (though the other 3 are). I could imagine a case where there is a valid n*n containing no balanced (n-1)*(n-1) grids.
That's an interesting question. If the grid is balanced, can you always find a balanced subgrid of any size. Maybe some type of argument with mean value theorem would work, although this is discrete not continuous
@@MrDowntownjbrown You don't have to go very far to find a counterexample:
[1 1 1]
[1 2 1]
[2 2 2]
If there is no balanced solution (or "as near to balanced as possible" for odd n), there cannot be an unbalanced solution either, because you have to add more tiles of a color that already has some squares.
@@screwaccountnames I believed you for a second, but I'm not sure I'm convinced. Suppose (wlog) there is a balanced 7x7 grid with 24 white tiles and 25 black tiles. Suppose that this grid contains a single square of 4 white tiles. One could substitute a single white tile for a black tile and potentially remove that square, leading to a 7x7 grid with 23 white and 26 black tiles, i.e. a valid but unbalanced grid.
I picked this option, because it looked like a smiley face
ｘｘｘｘｘ
ｘ ｘ
ｘ ｘ ｘ
ｘｘｘ
Alcesmire If you ask me, this deserves just as many bonus points as a Lego solution!
I made my life easy by using a randomized algorithm which simply generates random placements until one without squares is found. Worked very well for 5x5 at least.
Lego gets an honourable mention?
Well guess I’ll need to attempt the next one in Minecraft!
Please do!
"Sean M programmed this solution in Minecraft command blocks"
Or we could wire up a contraption using hoppers, comparators and locked repeaters.
There were at least two square in the Lego one though :')
I wrote a program (in Haskell) that would've found all of them... it was just really, really slow. I wanted to try and optimize it a bit first, then send it in, but I forgot.
It first calculated all of the potential squares for that size, then generated every possible game (with a balanced number of stones), then checked all of the games against those potential squares, filtered out those that had none, and printed them out. It spit out a size 5 solution every few seconds (after you gave it a while to get going), but it never got a size 6 solution. I gave it like 30 minutes.
I like to play the game "avoid the perfect square", also known by its other name "construct the parker square". very much fun :)
lol there's a square in the lego one (a small right windmill). I think Matt wanted us to see mistakes in squares to have us forget about his
Also an empty (green) square!
So, I entered this one, and did it free-hand right on the submission, didn't write anything down, just winged it, thought I had checked my answer thoroughly, and would be interested in seeing what I put in, after this solutions video came out so I could see where I went wrong, as it has been a little while... but I can't as far as I can tell.
The only problem I have with the puzzles us that theirs not enough of them. Keep it up, I love these videos.
Replacing trendy "hip" by "square" is groovy!
My aunt's just had a square replacement
7:07 It's Wookash
Good ol' Polish: writing a Slavic language with the Latin alphabet, and using none of the conventions of either.
Matt's little rectangle needs sound effects as it is driving around tje screen.
"This person did it with [insert programming language]" says nothing at all about how it was solved.
The 24 solutions for 6x6 looks suspiciously like the 18v18 case with no duplicate-cancellation, ignoring the 16 solutions for 17v19. In which case, the 0 solutions for the 7x7 board isn't conclusive for the 8x8 board - there being no 24v25 solutions doesn't immediately guarantee the absence of 23v26 solutions (it may be possible to prove that any solution with unequal numbers of the two colours can be turned into another solution by flipping a token of the majority colour, but it's not obvious that it's even true, let alone provable). So it would need to be shown in some way that you couldn't get a balanced 8x8 solution (32v32) by overlapping unbalanced (23v26 or more extreme) 7x7 solutions.
Presumably at some point the situation arises where you can't fit enough of one colour onto an NxN board even before considering the second colour - if you could prove that you can't fit 25 counters on a 7x7 in a square-free fashion, that would immediately prove that 7x7 and all larger boards are impossible.
The puzzle was to fill a 5x5 grid and form no squares. A solution confirms that cooperating players can engineer a tie game. But can a tie come about in normal play? In other words, does neither player have a winning strategy?
Steve the Cat Couch If my computer analysis is correct, the player who goes second can force a win; i.e., they can force the first player to eventually make a square. First player’s best hope is to make his first move to a corner. If second player does not also play to a corner, first player can then force a tie.
@@tomkeith8853 careful! If the second player does not play to an adjacent corner, it is a tie!
I love solutions. They are the best.
I love puzzles. They are better.
11:07 Thanks for the honourable mention, even though my solution was faulty! This is the square I missed: flic.kr/p/2jneLin
The Lego one has two small right windmills
I vote that LEGO should be worth a bonus point!
The "windmills" (2:27) made me think I might not have submitted a valid answer - I had checked for squares where the corners are "knight move" away from each other (the small windmills), but I don't remember checking for the large windmills. Fortunately, according to the Java widget and spreadsheet, my submission WAS a valid answer.
As I stated in the puzzle video - I conjecture that a g(64) by g(64) grid will never have a draw in the game of hip.
GNU Ron Graham.
the entire universe would collapse before the game ends. surely that counts as a draw.
Łucasz is pronounced Woo-cash
Whatever happened to the scrabble solution?
Thanks for the mention!
Thanks for doing the maths!
I wonder what the solutions are for avoiding Equilateral Triangles on a Triangular/Hex grid.
8:56 typo, 5x5 is 2704. I also got 24 for 6x6. I'm not sure what Matt means by duplicates?
He means reflections and rotations. A valid solution could be identical to another except they are rotated 90 degrees. Many people would prefer to remove those "duplicates" as they don't bring a new unique answer, just the same logic turned on its head.
@@rilesmcgiles1145 I see. He said "there *are* fewer than 24" which I thought implied there's no correct way to count 24, and they must've counted the same square twice.
@@aaronthompson2827 I got 24 as well including both symmetries and swapping the counter colours round. Matt might have been counting solutions that just exchange counter colours as duplicates. Similarly, the 2704 for 5x5 includes symmetric but not solutions that have 12 green counters as opposed to 13 green (or vice versa) - including both give 5408 for 5x5
@@rilesmcgiles1145 If a solution could be rotated by 90 degrees, it would have to have a lot of squares. Of course, reflections and 180-degree rotations are still valid.
I knew my messing about with Excel at age 6 in ICT was worth it.
I was watching a calculator unboxing video when this popped up in my subscriptions
You could do a single bonus point for those who do in novel ways
(a single bonus point is fun as an honorable mention it but not really worth much)
The trivial point of recognition.
I had not considered the windmill squares, but I got it right anyway.
Same, except I failed on a small windmill, sadly.
4:13 So is 340 the exact number of solutions for 5x5 if we consider reflections and rotations duplicates as one ?
didnt even know windmill was a type of square i was supposed to avoid angwy face
Dammit, I didn't think about the windmills.
Has Matt ever published the results for the Scrabble puzzle yet? I still got it wrong, and still have no idea why, and really want to know!!
No, we're all still waiting on it
Nope, and that's the only one I've gotten wrong so far, so I really want to see what the right answer is. (But I misplaced my answer, so I guess I won't really be able to see how close I was. Maybe I'll try again.)
No, an 8 by 8 does not necessarily contain a 7 by 7.
For a 7 by 7 grid, there are 49 cells and each player must thus make 24 or 25 moves.
An 8 by 8 grid has 64 squares and thus each player must make 32 moves, however 15 of these can be by one player along the 2 edges. In fact, they could place 28 squares all around the edge.
Thus any corner of the grid to make a 7 by 7 grid will have only 17 of their squares while it has 32 of the others players squares.
Sure, this wont produce a valid solution, but that doesn't mean that it is a trivial case of an 8 by 8 must have a 7 by 7 in it.
Now, how many solutions are there for a 7x7 BUT, there are 3 colors of tokens now. Now how big can you make the grid. and what if there are 4 players, or 5. If we take the sequence of the largest n×n square possible for 1, 2, 3, ... amount of tokens: {n_1, n_2, n_3 ...} what is that sequence? (we know it starts with i=1, n=1. then i=2, n=6, {1, 6, ...} but then?)
I only found 5 solutions for 5x5 by myself, and I was happy about that until I know that there are 2704 solutions in total. 😥
Idk why but i just completely forgot to check for the Small Windmills entirely. I almost got them all covered up by accident too even. It's a shame too, because I remember having "extra" tokens left and thinking, "aha, the trick is that you HAVE to place these extras and they might be forced to make a square. But sadly, no..
That Lego picture had a small right windmill in it.
Lego had a windmill in it
And an empty square
10:28, I believe you meant 5 x n, not 5 x 5.
I noticed what he calls the windmills while solving, but somehow I calculated them to be rhombuses and my submission included them 😒😒
Btw, what is the formula for weighted points ? I got 2000 on puzzle 9 and 500 (I think) on this, and my weighted score is 2100. Apparently the weight of the most recent is 100% and the immediate previous puzzle is 80%, but I don't have enough data to confirm
Lego should get all the bonus points.
8:43 The 5x5 has 2704 solutions, not 2074 but I assume that's not a math error..
Damn Matt, I was really hopeful I'd get included in the video.. I'll try again maybe..
Keep trying!
Is that Danielle Barker the Scrabble player?!
Hi
yes it's me, Hi Alec!
But they didn't call them hipsters, there were hippies, but then I'm not sure it was the hippies that used the word 'square' in this manner exclusively, there were also the beatniks, and other subcultures
Came to check out the sheet acid tabs in the thumbnail...
What if there are three players?
Wow! 5Xinfinity nice... I like the idea of toy LEGO Danish nationalism or whatever hahaha... anyway playing the game against someone with some weird set of rules on the 5Xinfinite board would be interesting perhaps to pursue with game theory etc
if Łukasz is Polish (which he probably is) it's pronounced like "wukass". (the L with stroke is a /w/ sound)
...and sz is close to a sh sound
LOL, I got 500 points and thought that meant I solved the puzzle. I don't even know what I did wrong. I think I didn't think about squares at angles, that screwed me over.
Shucks, I didn't even consider the windmill type squares
4:07 no-one tell him facecams are in the corner generally
The windmills got me...
Me too :'(
Ironic that the number of square-free solutions for 5×5 is 52 squared.
New puzzle suggestion:
How many different ways can Matt (mis)pronounce someone's name ;)
I don't think it was clear (or I want paying attention) that those other "squares" were to be avoided. I certainly didn't look. Shucks
I tried for like 5 minutes in excel, no luck. Should have tried lego.
smh didn't mention my program that requires 2.5TB of RAM
How did I end up here.
Dang, I got 500 points, I must have overlooked a square! :-(
How do you check your points? (I don't do facebook, twitter, or anything like that).
@@leefisher6366 Follow the "All details about Matt Parker's Maths Puzzles are on the Think Maths website" link in the description. Click on the "League Table And Scoring System" link. In the "Search by name..." field in the table, type the name you used in the submission form for the puzzle. The points are in the given week column, but only the past 6 weeks contribute to your total, so earlier weeks are not displayed.
@@msclrhd Thanks. (goes to check)
Yeah, I got 500 points too. Drat.
Gah! I wish I could see what it was I missed. I was sure I got all of them. I even went through and looked for every square from each coin that could make that square.
A lot of people missed squares formed by the other player’s tokens (aka: the blank squares) which would mean it is not a draw.
93 like and 930 views, nice!
I just noticed that Matt only uses the pronouns "they, them, their" to refer to participants.
Makes sense, since he can't really know their gender, but he also does it with his assistants, which is strange (did his assistants all ask for they pronouns? Unlikely)
Oliver is obviously male though. Why not just say “he”?
Yeah it's kind of annoying.
It's honestly nice, in situations where the gender of someone is useless information (most times), especially when the person is someone you don't personally know, I much prefer they pronouns. It's less presumptuous
Hey, he's a mathematician. If he can choose to use an abstraction that's always correct, instead of a guess that looks correct but has hidden exceptions, you know he's going to go for it.
Flood of Lego submissions incoming.
I fear as much.
So 2 minutes in and I'm 2nd comment
Woohoo!