আমি কন্টেস্ট মিস করায় ওস্তাদ। তবে গতকালকের ( আমি যেদিন লিখলাম,তার আগের দিনের) প্রবলেমগুলা চেক করার সময়ে Div-1 এর C/Div-2 এর A প্রবটা সল্ভ করার পরে খুব আনন্দ লাগে। :v এমন না সমস্যাটা অনেক কঠিন কিছু ছিলো,কিন্তু এর সমাধানের প্রক্রিয়াটা খুবই সুন্দর,তাই আজ এই সমস্যাটির সমাধান নিয়ে লিখবো।
সমস্যাঃ একটি ষড়ভুজ-এর ছয়টি বাহুর দৈর্ঘ্য দেয়া থাকবে। ষড়ভুজের ভিতর কয়টা সমান ক্ষেত্রফলের ত্রিভুজ আছে,সেটি বের করতে হবে। মজার ব্যাপার হচ্ছে,এটা তোমাকে নিশ্চিত করে দেয়া হয়েছে যে ষড়ভুজের অন্তঃস্থ সবগুলো কোণের মানই ১২০ ডিগ্রী! 😀 এই হিন্টসটার জন্যই সমস্যাটা এতোটা মজার! এবং ত্রিভুজগুলোর বাহুর দৈর্ঘ্য হবে ১ একক।
ধরি,একটা ষড়ভুজের ছয়টা বাহুর দৈর্ঘ্য হচ্ছে 1,1,1,1,1,1 ( সবগুলাই এক)
তাহলে এটি দেখতে কেমন হবে?
এমনঃ
চিত্রটায় ভালোভাবেই দেখা যাচ্ছে যে , সমবাহু ত্রিভুজের সংখ্যা হচ্ছে ৬ টি। আরেকটি উদাহরণ দেখে আসা যাক!
এবার আমরা ধরে নিই,বাহুগুলো হচ্ছে 1,2,1,2,1,2
তাহলে এটি দেখতে কেমন হবে?
গুণে গুণে দেখো,ত্রিভুজ আছে ১৩টা! 😀
তো তোমার কাজ এটাই,৬টা বাহুর দৈর্ঘ্য বলে দিবে,তোমাকে বলতে হবে এই ষড়ভুজটি সর্বোচ্চ কয়টি সমান ক্ষেত্রফলের ত্রিভুজ ধারণ করতে পারে!
সমাধানঃ
আমাদের আসল উদ্দেশ্য হচ্ছে,এই যে ষড়ভুজটি গঠিত হচ্ছে,প্রথমে এই ব্যাটার ক্ষেত্রফলটা কতো হবে,সেটা বের করে ফেলা! তারপরে আমরা ধরে নিবো এর ভিতরে 1 একক length এর n-সংখ্যক সমবাহু ত্রিভুজ আছে,তারপরে সেই n এর মান কতো সেটা বের করে ফেলবো,এবং এটিই হচ্ছে সমাধান! :3
প্রথমে উদ্দিষ্ট গুণাবলীর একটা ষড়ভুজ আঁকি! ^_^
এই হচ্ছে সেই গুণাবলীর একটা ষড়ভুজ,যার ক্ষেত্রফল বের করতে হবে। এর বাহুগুলোর নাম দেয়া উচিত! :/ না হয় দেখতে ভালো লাগছে না।
এবার সুন্দর দেখাচ্ছে। আমরা এবার এই ষড়ভুজটার বাহুগুলো বর্ধিত করে একে একটি সমবাহু ত্রিভুজে রুপান্তর করে ফেলবো। ছবি দেখিঃ
বাহুগুলোর নাম দিয়ে দিয়েছি,সাথে কোন বাহুগুলোকে বর্ধিত করে আমরা বড় সমবাহু ত্রিভুজের বাহুগুলো পেয়েছি,তার নামগুলোও দিয়ে দিয়েছি। আসলে চিত্রটা ভালোভাবে দেখলেই সব পানির মতো পরিস্কার হয়ে যাওয়ার কথা,সহজ জিনিস। চিত্র থেকেই দেখা যাচ্ছে যেঃ
a1 + a5 + a6 = a3 + a4 + a5 = a1 + a2 + a3
দেখতে সুন্দর লাগতেছে a1 , a2 আর a3 এর কম্বিনেশনটাই,তাই একে পছন্দ করলাম। :p
তাহলে এটিই হচ্ছে বড় সমবাহু ত্রিভুজটির একটি বাহু। আমরা জানি,একটি সমবাহু ত্রিভুজের ক্ষেত্রফলের সুত্র হচ্ছেঃ
Area = √3/4 * (বাহু)2
সুতরাং,বড় ত্রিভুজটির ক্ষেত্রফল হবেঃ
Area = √3/4 * (a1 + a2 + a3)2
এই বড় ত্রিভুজটি থেকে যদি আমরা এখন ডট ডট দেয়া অতিরিক্ত ছোট ছোট ত্রিভুজ তিনটি বাদ দিয়ে দিই,তাহলেই কি আমরা আমাদের নির্ধারিত ষড়ভুজটির ক্ষেত্রফল পেয়ে যাবো না?? 😀 অবশ্যই পেয়ে যাবো,চলো বাদ দিই!
Area of the Hexagon = √3/4 *{ (a1 + a2 + a3)2 – a12 – a32 – a52 } (কমন নিয়ে আর কি)
পেয়ে গেলাম আমাদের ষড়ভুজ-এর ক্ষেত্রফল! ^_^ এবার আমাদের কাজ হচ্ছে,এই ষড়ভুজটি কয়টা এক একক দৈর্ঘ্যের ছোট ছোট সমবাহু ত্রিভুজ আঁটতে পারে,তা খুঁজে বের করে প্রিন্ট করে দেয়া! 😀
ধরে নিই,ছোট ছোট n-সংখ্যক এক একক দৈর্ঘ্যের সমবাহু ত্রিভুজ মিলে ষড়ভুজটি গঠন করেছে,যাদের সম্মিলিত ক্ষেত্রফল খুব স্বাভাবিকভাবেই বড় ষড়ভুজটির সমান। তাহলে লিখা যায়ঃ
এই সহজ সমীকরণটি থেকে অনেকটা কোনো ক্যালকুলেশন না করেই বলে দেয়া যায়,n সমান সমান কতো!
এই n ই হচ্ছে নির্দিষ্ট সর্বোচ্চ সংখ্যক সমবাহু ত্রিভুজের সংখ্যা! 🙂
তোমরা সমস্যাটির সমাধান তো পেয়ে গেলে,এবার নিজ নিজ ভাষায় কোড করে সাবমিট করে দেখে আসো! আমি সমস্যাটির লিংক দিয়ে দিচ্ছিঃ
Gerald’s Hexagon
বিদায়!
অসাধারণ!
ধন্যবাদ!
বস 8|
Thanks for the post. I am in a confusion. If the angles were not equal i.e, instead if 120 degree, they vary each other. Then what ? Will this tutorial be okay for that case? Thanks vaiya.
এখানে a1 বাহু বিশিষ্ট সমবাহু ত্রিভুজের প্রতিটি কোণের মান 60 ডিগ্রি । একই কথা a3 ও a5 বাহু বিশিষ্ট সমবাহু ত্রিভুজের বেলাও । সুতরাং বড় ত্রিভুজটিও সমবাহু [যেহেতু সমবাহু ত্রিভুজের রতিটি কোণের মান 60 ডিগ্রি ] এজন্যই a1 + a5 + a6 = a3 + a4 + a5 = a1 + a2 + a3 [ত্রিভুজের সমান সমান কোণের বিপরীত বাহুদ্বয় সমান] । কাজেই পদ্ধতিটি ঠিক আছে এবং কাজ করবে বিষম এবং সুষম উভয় বহুভুজের ক্ষেত্রফল নির্ণয়ের বেলায় ।
ষড়ভুজটি সুষম না হলে এই সূত্র খাটবেনা,ধন্যবাদ।
Oawww!!!!
খুবই ভালো । বিষম বহুভুজের পাশাপাশি বাহুর কোণ বের করার কোন পদ্ধতি আছে ?
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Eternity Puzzle is a nifty utility designed for playing a short game resembling Connect Four. The puzzle consists of 8 rows of 16 moves each, which are horizontally or vertically divided into identical blocks of 4 moves. The first block is black and the second is white, and each player can manipulate 4 moves per turn. The first game is of the first player starting in the upper left corner and the second in the upper right corner.
The application can draw an image on a specified shape. http://maps.google.hn/url?q=https://greserovig.weebly.com
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Pros
Gives you an overview of your most important SEO, speed, security-related practices, and guidelines
Very easy to use
Has a polished GUI
Reproduces audit results in an intuitive and visually pleasing manner
Cons
Hotspot protected, no options for resolution
Multi-user access not supported
We’d almost be finished with our review, however, there’s one more thing you should consider.
You can grab a one-month free trial version of Checkbot https://gastluaquacon.weebly.com
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Bazinga4PC is a handy and reliable utility designed to convert any web site into a Windows program.
This allows you to access your favorite web sites quickly and easily without relying on your web browser which could crash at any time. Bazinga4PC is designed for web masters to push web apps to the Windows Desktop.
Bazinga4PC is a handy and reliable utility designed to convert any web site into a Windows program.
This allows you to access your https://rirerapca.weebly.com
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Read more at Blogdex.com about apps blog posts
CMS is a web application used to interact with all kinds of websites and it is flexible enough to easily adjust its design according to the order of its content. CMS functions, in general, are divided in three main parts: the Content Management System, the user interface and the database. The three of them are very closely connected to each other, as the various modules they are composed of have to be connected to the other by https://unalgedtio.weebly.com
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