Solution to the four-colour problem in the movie X+Y @bpolster

bpolster Thank you to everybody who contributed a solution to this problem in the comments section of the main video:

youtu.be/04x4ZdLpN-0

What I like about the solution I present in this video is that it also gives a complete characterization of these special colorings of square grids on top of giving the answer to the problem.

Lots of other nice approaches to this problem are possible. I'd just like to mention one more which I particularly like, not least because it uses a very cute trick that I also talk about in this video

youtu.be/9JN5f7_3YmQ

Basically you prove that you get different color corners in any 4-colouring of a 4x4 grid. Then you scale up the proof as indicated in this diagram

imgur.com/a/affkdH7

So for a special coloring of the 8x8 grid you divide the grid into four 4x4s, and combine the corners of these four 4x4 into another 4x4. Now it's not terribly hard to see that this new 4x4 is also colored in a special way. Therefore its corners, which are also the corners of the 8x8 are have to be of different color. Nice, and in the next step divide a 16x16 into four 8x8s, etc.

Some of you actually argued in this way. The only little problem was that most of you who did argue this way thought that it was completely obvious that the new 4x4 carries a special coloring, which is not the case. Remember that you have to check that all 2x2s in this new 4x4 are colored differently. This is really obvious for most of these 2x2s but not for all of them. In particular for the 2x2 consisting of the squares 2,5,4,7 in my diagram you really have to argue separately that these are of different color (e.g. like I did in this video).

Anyway, who gets the t-shirt? Bit tricky. The first submission was this:

Joeeeee For the challenge: Same as Pascallian Triangles but for n=4. youtu.be/9JN5f7_3YmQ Where can I get my t-shirt? 😜

So that looks like Joeeeee is on the right track and that the proof is supposed to be something like the one I just outline. A "bit" short on details though :) The next proof that was submitted and that I thought worked was also along the same lines. This proof was by Tommaso Gianiroio. Again not quite complete, but complete enough to warrant a t-shirt :)

Anyway, I am in a generous mood today and so happy to give a t-shirt to both Joeeeee an Tommaso. Please get in touch with me through a comment on this video and by e-mail.

updated 2 years ago

Solution to the four-colour problem in the movie X+Y

bpolster 2021-10-21 | Thank you to everybody who contributed a solution to this problem in the comments section of the main video:

youtu.be/04x4ZdLpN-0

What I like about the solution I present in this video is that it also gives a complete characterization of these special colorings of square grids on top of giving the answer to the problem.

Lots of other nice approaches to this problem are possible. I'd just like to mention one more which I particularly like, not least because it uses a very cute trick that I also talk about in this video

youtu.be/9JN5f7_3YmQ

Basically you prove that you get different color corners in any 4-colouring of a 4x4 grid. Then you scale up the proof as indicated in this diagram

imgur.com/a/affkdH7

So for a special coloring of the 8x8 grid you divide the grid into four 4x4s, and combine the corners of these four 4x4 into another 4x4. Now it's not terribly hard to see that this new 4x4 is also colored in a special way. Therefore its corners, which are also the corners of the 8x8 are have to be of different color. Nice, and in the next step divide a 16x16 into four 8x8s, etc.

Some of you actually argued in this way. The only little problem was that most of you who did argue this way thought that it was completely obvious that the new 4x4 carries a special coloring, which is not the case. Remember that you have to check that all 2x2s in this new 4x4 are colored differently. This is really obvious for most of these 2x2s but not for all of them. In particular for the 2x2 consisting of the squares 2,5,4,7 in my diagram you really have to argue separately that these are of different color (e.g. like I did in this video).

Anyway, who gets the t-shirt? Bit tricky. The first submission was this:

Joeeeee For the challenge: Same as Pascallian Triangles but for n=4. youtu.be/9JN5f7_3YmQ Where can I get my t-shirt? 😜

So that looks like Joeeeee is on the right track and that the proof is supposed to be something like the one I just outline. A "bit" short on details though :) The next proof that was submitted and that I thought worked was also along the same lines. This proof was by Tommaso Gianiroio. Again not quite complete, but complete enough to warrant a t-shirt :)

Anyway, I am in a generous mood today and so happy to give a t-shirt to both Joeeeee an Tommaso. Please get in touch with me through a comment on this video and by e-mail.

test2blank #shorts

bpolster 2021-12-23 | ...

blank test #shorts

bpolster 2021-12-23 | ...

viviani test 2 #shorts

bpolster 2021-12-23 | ...

viviani #shorts

bpolster 2021-12-22 | ...

Pachabels Canon in D upside down in reverse direction

bpolster 2010-06-04 | This is what a music roll of the canon in D sounds when you play ii upside down and in reverse direction. We also played this roll the right side up, upside down, the right side up in reverse direction (see my channel).

At this year's Gathering for Gardner, Vi Hart presented a couple of similar experiments. The two that I liked best were Canon in D played on four music boxes (http://www.youtube.com/watch?v=3a9wWRxYSko) and a music box playing a piece recorded on a paper strip that was twisted into a Mobius band (http://www.youtube.com/watch?v=3iMI_uOM_fY).

mathematical magic trick 3

bpolster 2010-06-04 | Lately we've been discussing mathematical card tricks as part of the maths circle at the John Monash Science School here in Melbourne. I usually demonstrate a trick and then ask the students to figure out how it works. In the first instance these videos are meant for the students who attend these meetings. If you are interested in mathematical card tricks I recommend the MAA column www.maa.org/columns/colm/cardcolm.html

Pachabels Canon in D

bpolster 2010-06-03 | This is a music roll of the canon in D played on our player piano. We also played this roll upside down, the right side up in reverse direction, and upside down and in reverse direction (see my channel).

At this year's Gathering for Gardner, Vi Hart presented a couple of similar experiments. The two that I liked best were Canon in D played on four music boxes (http://www.youtube.com/watch?v=3a9wWRxYSko) and a music box playing a piece recorded on a paper strip that was twisted into a Mobius band (http://www.youtube.com/watch?v=3iMI_uOM_fY).

Pachabels Canon in D upside down

bpolster 2010-06-03 | This is what a music roll of the canon in D sounds when you play it upside down. We also played this roll the right side up, the right side up in reverse direction, and upside down and in reverse direction (see my channel).

At this year's Gathering for Gardner, Vi Hart presented a couple of similar experiments. The two that I liked best were Canon in D played on four music boxes (http://www.youtube.com/watch?v=3a9wWRxYSko) and a music box playing a piece recorded on a paper strip that was twisted into a Mobius band (http://www.youtube.com/watch?v=3iMI_uOM_fY).

juggling the changes

bpolster 2010-06-03 | To really be able to appreciate what this video is all about you need to know what it means to Ring the Changes. What I am doing here is to juggle a full extent on four bells in the following sense: the different bells are associated with the different bells and whenever a bell rings I toss its associated ball in the air. Just in case you are wondering, the extent in question is not Plain Bob. I made up this video as a fun ending to the following article on the math of ringing the changes http://plus.maths.org/issue53/features/polsteross/index.html

mathematical magic trick 2 (kruskal card trick with dice)

bpolster 2010-06-03 | Lately we've been discussing mathematical magic tricks as part of the maths circle at the John Monash Science School here in Melbourne. I usually demonstrate a trick and then ask the students to figure out how it works. In the first instance these videos are meant for the students who attend these meetings. Here is the trick in its original card setting http://www.youtube.com/watch?v=kvvq0ERlixU&feature=related

mathematical magic trick 1

bpolster 2010-06-03 | Lately we've been discussing mathematical card tricks as part of the maths circle at the John Monash Science School here in Melbourne. I usually demonstrate a trick and then ask the students to figure out how it works. In the first instance these videos are meant for the students who attend these meetings. If you are interested in mathematical card tricks I recommend the MAA column www.maa.org/columns/colm/cardcolm.html

Playing the knotted didgeridoo

bpolster 2010-04-10 | In this video I invite you to compare the sound of a proper didgeridoo made from wood and that of a knotted didgeridoo made from twelve PVC drainage pipe pieces. This video is part of a gift exchange that I prepared for this year's Gathering for Gardner (http://g4g4.com)