tag:blogger.com,1999:blog-5303307482158922565.post168216483944085790..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: Tin Ceilings, Triangles, and Loving MathSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger33125tag:blogger.com,1999:blog-5303307482158922565.post-67304262919703579492015-02-27T20:56:11.736-08:002015-02-27T20:56:11.736-08:00I'm currently taking a course on programming a...I'm currently taking a course on programming algorithms at SDSU and it is by far the most math-intensive computer science course there is. I think it'd be very useful to learn how to come up with a formula for a problem like you have. It seems like an essential skill for writing programs designed to calculate an answer for a problem.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-53118208119214841242014-12-22T07:46:29.440-08:002014-12-22T07:46:29.440-08:00Leanne, I'm not sure what you're asking. T...Leanne, I'm not sure what you're asking. There are 268 triangles in that 4x4 picture in the post. To find them takes a lot of work, and it's a lot more interesting if you're trying to find patterns, and understand them.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-3296133329226168892014-12-20T22:54:47.764-08:002014-12-20T22:54:47.764-08:00so what is the correct answer?so what is the correct answer?Leanne Barkerhttps://www.blogger.com/profile/17739794646818272799noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-23148891070570496592014-06-07T00:45:27.847-07:002014-06-07T00:45:27.847-07:00"I found one by starting in the top left corn..."I found one by starting in the top left corner of the 4x4"<br /><br />I meant top right corner.<br /><br />"That did seem odd, didn't it? (More 18s than 16s.)"<br /><br />Yes, I wasn't expecting that, which is why I was mostly focusing on finding more triangles smaller than 18.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-17336521019862594682014-06-06T21:28:03.341-07:002014-06-06T21:28:03.341-07:00That did seem odd, didn't it? (More 18s than 1...That did seem odd, didn't it? (More 18s than 16s.)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-68694763634034874402014-06-06T21:24:45.176-07:002014-06-06T21:24:45.176-07:00I finally found the last 4. There were 4 18s I mis...I finally found the last 4. There were 4 18s I missed. I found one by starting in the top left corner of the 4x4, shifting left 1 square, then shifting down 1 square. I forgot to check for this size triangle closer to the center of the 4x4. Also, I didn't think there would be more 18s than 16s since 18s are larger.<br /><br />My final count:<br /><br />64 size 1s<br />64 size 2s<br />48 size 4s<br />36 size 8s<br />24 size 9s<br />12 size 16s<br />16 size 18s<br /> 4 size 32s<br /><br />268 total.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-77325510908712764752014-06-06T00:51:23.291-07:002014-06-06T00:51:23.291-07:00Ok, I finally found 3 18s facing each direction, b...Ok, I finally found 3 18s facing each direction, bringing my total to 264. Counting all of the triangles in 1x1, 2x2, and 3x3 squares helped me find triangles in the 4x4 I previously missed.<br /><br />My revised count:<br /><br />64 size 1s<br />64 size 2s<br />48 size 4s<br />36 size 8s<br />24 size 9s<br />12 size 16s<br />12 size 18s<br /> 4 size 32s<br /><br />Only 4 more to find!Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-91278226982411533402014-06-06T00:32:17.494-07:002014-06-06T00:32:17.494-07:00Yes, I suppose explaining how you made your formul...Yes, I suppose explaining how you made your formula does serve as a proof.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-75798966241363638962014-06-05T21:46:23.577-07:002014-06-05T21:46:23.577-07:00Hmm, I didn't write my process out as a proof,...Hmm, I didn't write my process out as a proof, but I feel like I have proved it.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-37901108541004372552014-06-05T21:44:18.202-07:002014-06-05T21:44:18.202-07:00The math-future (or math 2.0) discussion is here: ...The math-future (or math 2.0) discussion is here: https://groups.google.com/forum/#!topic/mathfuture/W41lMxjhQ1USue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-36775102977386714092014-06-05T21:20:09.727-07:002014-06-05T21:20:09.727-07:00Yep, it was the rounding. I just entered it as i/2...Yep, it was the rounding. I just entered it as i/2. I didn't understand how to use the round function on the calculator, so I just set it up so that everything gets rounded to the closest integer. From the mode menu I changed the rounding from float to 0. After testing out the rounding, I learned that rational numbers with 5 at the end of the decimal portion are rounded up. So i/2 was still rounded to the next highest integer, but it may not have been the only thing that was rounded. I now understand how to use the round function: first operand is number to be rounded: second operand is number of decimal places. So I set the global rounding setting back to float and replaced i/2 with round(i/2), giving me the result 268. I'm still interested in seeing if I can find the last 8 triangles from examining the 4x4. As far as I know, there is no proof of the formula.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-55220310205955905162014-06-05T20:17:23.200-07:002014-06-05T20:17:23.200-07:00What did you do with their round(i/2)? They meant ...What did you do with their round(i/2)? They meant the same thing I did with ceiling(i/2), though it's not as obvious with the term round what should happen when you're halfway between two numbers. Unfortunately, I no longer have a TI-84. Perhaps it doesn't do round(i/2) the way the OEIS meant it. (I tried to google it, and couldn't figure it out for sure.) If you use int(i/2+.5), I think you'll get ceiling. int() is the equivalent of floor. (They may act differently on negatives, but that probably depends on what device you're using.)<br /><br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-33187890469672663732014-06-05T17:49:54.727-07:002014-06-05T17:49:54.727-07:00I just entered the summation form of the formula g...I just entered the summation form of the formula given on On-line Encyclopedia of Integer Sequences into my calculator (TI-84 plus) for a 4x4, and I got 280, the same answer posted by Cooper Macbeh in the Math 2.0 discussion. But when I enter the closed form I get 268. Why's that?Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-18072618695743862692014-06-05T12:27:53.960-07:002014-06-05T12:27:53.960-07:00Perhaps not as flawed as you're thinking. Each...Perhaps not as flawed as you're thinking. Each time we approach a problem a new way, or another time, it adds to what we've done before. Think of your "flawed" approach as getting to know the problem.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-17633873270642243872014-06-05T12:13:03.669-07:002014-06-05T12:13:03.669-07:00That's what I've been doing after realizin...That's what I've been doing after realizing how flawed my original approach was.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-39635236705924842432014-06-05T07:25:19.300-07:002014-06-05T07:25:19.300-07:00Sorry, I threw out my notes when I was cleaning up...Sorry, I threw out my notes when I was cleaning up last night. My recommendation is that you outline one and slowly shift it left or right up or down.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-8102332102186019562014-06-04T22:21:57.486-07:002014-06-04T22:21:57.486-07:00I just saw that there are three 16s facing each di...I just saw that there are three 16s facing each direction, so 12 total. But I'm still only seeing two 18s facing each corner. Now I've found 260 triangles.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-44377195800270623962014-06-04T22:01:56.813-07:002014-06-04T22:01:56.813-07:00I think you missed some 16s and 18s.I think you missed some 16s and 18s.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-15110013472684189232014-06-04T21:45:39.241-07:002014-06-04T21:45:39.241-07:00So here's my revised count:
256 total
64 si...So here's my revised count:<br /> <br />256 total<br /><br />64 size 1s<br />64 size 2s<br />48 size 4s<br />36 size 8s<br />24 size 9s<br />8 size 16s<br />8 size 18s<br />4 size 32s<br /><br />which 12 am I missing?Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-72564117305858938862014-06-04T21:38:36.074-07:002014-06-04T21:38:36.074-07:00>Wow, I'm surprised at the quick response.
...>Wow, I'm surprised at the quick response.<br /><br />Ahh, the miracles of modern technology.<br /><br />>I'm not sure what you mean by ceiling.<br /><br />It means, if your number isn't a whole number, round up to the whole number above.<br /><br />>this is going to be even harder than I thought.<br /><br />Sleep on it a few times. You'll get it.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-82824516492254535802014-06-04T20:35:43.405-07:002014-06-04T20:35:43.405-07:00Oh, I finally see the size 9s. Damn, this is going...Oh, I finally see the size 9s. Damn, this is going to be even harder than I thought.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-2840893373244806042014-06-04T20:20:35.937-07:002014-06-04T20:20:35.937-07:00I knew there must be someway to come up with a for...I knew there must be someway to come up with a formula for finding the total number of triangles, but I didn't even know how to start on that. Right now I'm trying to understand yours, but I'm not sure what you mean by ceiling.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-55410957636290107542014-06-04T20:13:46.811-07:002014-06-04T20:13:46.811-07:00Wow, I'm surprised at the quick response. You&...Wow, I'm surprised at the quick response. You're right, there are size 18s! How did I miss those? I was convinced that the different sizes of the triangles were all powers of 2. I counted 4 18s, bringing my total to 188. I don't see any size 9s, where are you seeing them? I don't think you can have triangles larger than very small that are made up of an odd number of very small triangles.Anonymoushttps://www.blogger.com/profile/17959687959176018305noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-47572298715503420542014-06-04T13:50:43.642-07:002014-06-04T13:50:43.642-07:00I just double-checked my 268. Hao, is that what yo...I just double-checked my 268. Hao, is that what you get? I get 492 for the 5x5 rectangle. And I'm working on the nxn...<br /><br />Evens and odds work a bit differently, but if I take a hint from Hao's floor, and use ceiling, which fits my thinking better, I get 4*sum(i=1 to n) of [(n+1-i)(2n+2-i-ceiling(i/2))]. That's the simplified form. It's easier to see the meaning before you algebraically simplify.<br /><br />The triangles that have a diagonal hypotenuse are easier to describe in terms of n. They point 4 directions, so it's 4 times something. The number you find depends on the size, and you'll get (n+1-i)^2 of these in each direction, so 4*(n+1-i)^2, for i going from 1 to n.<br /><br />Now for the triangles with a vertical or horizontal hypotenuse. Using hypotenuse on the bottom to think (knowing there will be 4 directions), we see the smallest ones go up 1/2 of a rectangle, the next size (which are 4 triangles) go up a whole rectangle, the next size (9 triangles) go up 1 1/2 rectangles. When we count how many of these across and how many rows going up, we need to round up those fractions. That's why I use ceiling. So for these I get 4*(n+1-i)(n+1-ceiling(i/2)).<br /><br />Adding the two parts together, we get 4*(n+1-i)^2 + 4*(n+1-i)(n+1-ceiling(i/2)), which simplifies to 4*(n+1-i)[2n+2-i-ceiling(i/2)]. <br /><br />I spent a very long time on this three years ago, and left it entirely until now. I am amazed that my brain held onto whatever it learned doing this, and I was able to solve it completely in an hour and a half just now. Very interesting. (I think of myself as having a very bad memory, but the part of my memory that holds the thinking process for this works just fine!)<br /><br /><br />I finally went and checked <a href="http://divisbyzero.com/2011/02/10/counting-triangles-on-a-tin-ceiling-solution-2/" rel="nofollow">Dave Richeson's solution</a>. He approaches it quite a bit differently. I checked my answer against his, saw I had a mistake in the formula and fixed it. He simplifies further, since summations can always be re-written as formulas without summation. He also finds the sequence of numbers we get, 8,44,124,268,492,... for rectangles of side length 1,2,3,... in the <a href="http://oeis.org/search?q=8%2C44%2C124%2C268&language=english&go=Search" rel="nofollow">On-line Encyclopedia of Integer Sequences</a>. Very cool!<br /><br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-62480896877660191822014-06-04T11:41:50.073-07:002014-06-04T11:41:50.073-07:00CMD619, I get 268, but I still haven't solved ...CMD619, I get 268, but I still haven't solved for a rectangle of size nxn. <br /><br />I don't see enough different sizes in your list. You left out the 9s and 18s. Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.com