10567 Players currently online!
Man vs. Machine - good luck!
Turn-based games at any time!
Vote for the best move to win!
Do you have what it takes?
Sharpen your tactical vision!
Get advice and game insights!
Learn from top players & pros!
View millions of master games!
Your virtual chess coach!
Perfect your opening moves!
Test your skills vs. computer!
Find the right private coach!
Can you solve it each day?
Bring it all together!
Beginners, start here!
Make friends & play team games!
News from the world of chess!
Search all Chess.com members!
Find local clubs & events!
Who's the best of your friends?
Read what members are saying!
I originally posted this in a chess.com group, but since it is a small group, I thought more people might appreciate it here.
I made up both of these problems on my sabbath, when I can't write/use a calculator, so both of them can (and should!) be done in one's head. Scroll down for the solutions.
1. The CIA is using sensory deprivation chambers as a new form of "enhanced" interrogation. They want you to tell them what the area of a triangle is if the triangle has side lengths of 5 in, 9 in, and sqrt(34) in. Since you are in a sensory deprivation chamber, you cannot use writing materials, counting materials, or a calculator. In addition, it is against your unbreakable moral principles to use Heron's formula. What is the area of the triangle?
2. There is a 20% probability that an event will occur at least once in a year. What is the approximate probability (accurate to 1/2 a percent) that the event will occur at least once in 10 years? The hard part about this problem is probably doing it in one's head.
Envision the triangle above. You need to solve for h so as to use the formula A = 1/2bh. From the above diagram, you can derive the system of equations x + y = 9; x^2 + h^2 = 25; y^2 + h^2 = 34. Fortunately, it is not particularly difficult to solve this system for h in one's head, since many squared variables "drop out." x + y = 9 so x = 9 - y. Substitute that into x^2 + h^2 = 25 and you get (9 - y)^2 + h^2 = 25 so y^2 - 18y + 81 + h^2 = 25 so y^2 + h^2 - 18y + 56 = 0. The third equation, y^2 + h^2 = 34, can be changed to y^2 + h^2 - 34 = 0 and then substituted into y^2 + h^2 - 18y + 56 = 0 to get y^2 + h^2 - 18y + 56 = y^2 + h^2 -34 so -18y + 56 = -34 so -18y = -90 so y = 5. Substitute that into y^2 + h^2 = 34 to get 25 + h^2 = 34 so h^2 = 9 so h = 3. A = 1/2bh = 1/2(9)3 = 13.5 sq. in.
2. P(event occurs at least once) = 1 - P(event does not occur) = 1 - 0.8^10 = 1 - (4/5)^10 = 1 - 4^10/5^10 = 1 - (2 * 2)^10/(10/2)^10 = 1 - (2^10 * 2^10)/(10^10/2^10) = 1 - (1024 * 1024)/(10^10/1024) = 1 - (a little more than 10^6)/(a little less than 10^7) = 1 - a little more than 0.1 = a little less than 0.9. The probability that the event will occur at least once in 10 years is about 89% (the probability as determined by a calculator is 89.26258176...%).
Don't be intimidated by the number of equations in each of the solutions; trust me, it really isn't difficult to do in one's head. I carefully explained the solution to #1 orally to a pretty good math student, and he was, with some thought, then able to solve for a the area of a different triangle in his head, given their side lengths. I actually got the idea for #2 when I was reading an article that predicted the probability of a nuclear attack against a major city occurring in any given year to be 20% (scary thought! I'm hoping that's an overestimate!).
Well yes you're right. I've only studied basic trig, so I wouldn't know... but could this be a proof for the law of cosines? I've tried to turn it into a proof for Heron's formula, and it gets messy... really messy... it makes me make squealing noises.
by GuessWhoIAm 5 minutes ago
Stuff Non-Chess Players Say
by GMrisingJCLmember1 5 minutes ago
Chance to play a NM!
by Eseles 9 minutes ago
What's your favorite excuse for why you lost a game of chess?
by denniscomeaux33 13 minutes ago
eliminating blunders without the wait?
Has my game improved? Beat 1600
by Blockah 17 minutes ago
Chess Puzzle From A Real Game
by ponz111 25 minutes ago
who do you think is the best chess player ever?
by JustinJ_FairfieldU 28 minutes ago
I Like Carlsen !
by cabadenwurt 29 minutes ago
Moving a Pawn Twice in the Opening
by ponz111 29 minutes ago
Why Join | Chess Topics |
Help & Support |
© 2015 Chess.com
• Chess - English
We are working hard to make Chess.com available in over 70 languages. Check back over the year as we develop the technology to add more, and we will try our best to notify you when your language is ready for translating!