Comments for Tanya Khovanova's Math Blog http://blog.tanyakhovanova.com Mathematics, applications of mathematics to life in general, and my life as a mathematician Sun, 19 May 2013 04:32:23 +0000 http://wordpress.org/?v=2.3 Comment on Happy Nobel Prize Winners by Bingo Crepiscule http://blog.tanyakhovanova.com/?p=455#comment-317331 Bingo Crepiscule Thu, 16 May 2013 07:52:43 +0000 http://blog.tanyakhovanova.com/?p=455#comment-317331 The Physics prizes were awarded to 54 phenomenally young scientists in the period 1901-1950. I calculated that the average age was 46. The average ages of the prize winners in Chemistry, Medicine, Literature and Peace were significantly higher. I guess if this analysis was performed on the Physics prize winners and nominees only, in the first half of the last century, we may begin to make a better conclusion, albeit with a small data set. The Physics prizes were awarded to 54 phenomenally young scientists in the period 1901-1950. I calculated that the average age was 46. The average ages of the prize winners in Chemistry, Medicine, Literature and Peace were significantly higher. I guess if this analysis was performed on the Physics prize winners and nominees only, in the first half of the last century, we may begin to make a better conclusion, albeit with a small data set.

]]> Comment on Divisibility by 7 is a Walk on a Graph. II by Firoze http://blog.tanyakhovanova.com/?p=262#comment-316543 Firoze Tue, 14 May 2013 11:52:21 +0000 http://blog.tanyakhovanova.com/?p=262#comment-316543 Brilliant! Thank you! Brilliant! Thank you!

]]> Comment on Divisibility by 7 is a Walk on a Graph, by David Wilson by Firoze http://blog.tanyakhovanova.com/?p=159#comment-316540 Firoze Tue, 14 May 2013 11:50:32 +0000 http://blog.tanyakhovanova.com/?p=159#comment-316540 I am NOT AT ALL a mathematician... but this was just too cool!! Thank you so much for sharing. :) I am NOT AT ALL a mathematician… but this was just too cool!! Thank you so much for sharing. :)

]]> Comment on The Sleeping Beauty Problem by Ioannis Mariolis http://blog.tanyakhovanova.com/?p=356#comment-316080 Ioannis Mariolis Mon, 13 May 2013 09:27:56 +0000 http://blog.tanyakhovanova.com/?p=356#comment-316080 The probability of Heads should always be 1/2, since upon awakening no gain or loss of information takes place. The betting argument you have presented is erroneous. Let's assume even bets on Heads/Tails (e.g. in case of heads you win 600 $, in case of Tails you lose 600 $). Then, if you bet on Heads your expected gain according to halfers is E(Heads)=1/2*600+1/2(-1200)=-300 , which means that in fact you have an expected loss of 300 $. According to thirders your expected gain is E(Heads)=1/3*600+2/3(-600)=-200, which means that in fact you have an expected loss of 200 $. If you are not convinced that the halfers approach is the correct one you can run a large number of trials of the experiment and you will see it is the halfers approach that gives the correct prediction about your loss. In case the bet is not even, the correct threshold that should define your strategy is to take the bet if you win at least twice than what you lose. Coincidentally, this threshold is predicted by both approaches. The probability of Heads should always be 1/2, since upon awakening no gain or loss of information takes place. The betting argument you have presented is erroneous. Let’s assume even bets on Heads/Tails (e.g. in case of heads you win 600 $, in case of Tails you lose 600 $). Then, if you bet on Heads your expected gain according to halfers is E(Heads)=1/2*600+1/2(-1200)=-300 , which means that in fact you have an expected loss of 300 $. According to thirders your expected gain is E(Heads)=1/3*600+2/3(-600)=-200, which means that in fact you have an expected loss of 200 $. If you are not convinced that the halfers approach is the correct one you can run a large number of trials of the experiment and you will see it is the halfers approach that gives the correct prediction about your loss. In case the bet is not even, the correct threshold that should define your strategy is to take the bet if you win at least twice than what you lose. Coincidentally, this threshold is predicted by both approaches.

]]> Comment on A Problem from the Moscow Olympiad by Tanya Khovanova http://blog.tanyakhovanova.com/?p=456#comment-315882 Tanya Khovanova Sun, 12 May 2013 21:07:33 +0000 http://blog.tanyakhovanova.com/?p=456#comment-315882 Thank you, Vladimir, "common" should be removed. Thank you, Vladimir, “common” should be removed.

]]> Comment on A Problem from the Moscow Olympiad by Vladimir http://blog.tanyakhovanova.com/?p=456#comment-315816 Vladimir Sun, 12 May 2013 16:54:04 +0000 http://blog.tanyakhovanova.com/?p=456#comment-315816 >Prove that all the people have exactly the same number of common acquaintances at this meeting. Isn't "common" superfluous? >Prove that all the people have exactly the same number of common acquaintances at this meeting.
Isn’t “common” superfluous?

]]> Comment on My Weight by Olga http://blog.tanyakhovanova.com/?p=447#comment-315082 Olga Fri, 10 May 2013 18:44:53 +0000 http://blog.tanyakhovanova.com/?p=447#comment-315082 Здравствуйте, Таня! У меня была такая же проблема. В августе я весила 110 кг. Три месяца сидела на порошковой диете Modifast. Похудела на 20 кг. А в ноябре мне сделали операцию по уменьшению желудка. Сейчас уже вешу 75 кг. Вес уходит, хотя очень медленно. Есть могу почти все (кроме сахара), но понемногу. Но мне хватает. Очень сомневалась на счет операции, но в Швеции,где я живу, эти операции довольно распространены, и за последние годы довольно много знакомых сделали такие операции. Так что я рискнула, и не жалею Здравствуйте, Таня! У меня была такая же проблема. В августе я весила 110 кг. Три месяца сидела на порошковой диете Modifast. Похудела на 20 кг. А в ноябре мне сделали операцию по уменьшению желудка. Сейчас уже вешу 75 кг. Вес уходит, хотя очень медленно. Есть могу почти все (кроме сахара), но понемногу. Но мне хватает. Очень сомневалась на счет операции, но в Швеции,где я живу, эти операции довольно распространены, и за последние годы довольно много знакомых сделали такие операции. Так что я рискнула, и не жалею

]]> Comment on The Odd One Out by rp987 http://blog.tanyakhovanova.com/?p=182#comment-314290 rp987 Wed, 08 May 2013 17:07:31 +0000 http://blog.tanyakhovanova.com/?p=182#comment-314290 The leftmost red suqare with a black border. each of the rest would be considered odd one out aside from this square. 2nd - no border 3rd - Round 4th - Green 5th - Small by definition - they're all odd out aside from the first, making it the most odd among odds. The leftmost red suqare with a black border.

each of the rest would be considered odd one out aside from this square.

2nd - no border
3rd - Round
4th - Green
5th - Small

by definition - they’re all odd out aside from the first, making it the most odd among odds.

]]> Comment on One-Way Functions by Gary http://blog.tanyakhovanova.com/?p=277#comment-313001 Gary Sat, 04 May 2013 18:26:58 +0000 http://blog.tanyakhovanova.com/?p=277#comment-313001 So for the one way hash, where the original sequence cannot be (should not be) recoverable, I have literally used hash. The food is a combination of ingredients is specific proportions that produce a unique flavor once combined. Without all sorts of chemical analysis (the weakness) the original recipe cannot be discovered. All the ingredients are put into a grinder which homogenizes them into a single consistent product. Unable to pick out the original pieces this is hash! Now for math purposes it can be more specific where original text is mutated into something else that is simpler to manage and does not require as much protection. For example, if we do a simple hash algorithm which assigns all the letters of a password to a number and adds all those numbers together we get a number from which it is impossible to retrieve the original password (unless it was something simple or very repetitive). While that protect the original password, it has a weakness because many other passwords could produce the same number (collisions); "password" and "drowssap" would produce the same number as would "ssawprod" and many others. So the true objective of the math for a one way has is to generate a unique value and reduce (or eliminate) the potential of collisions. With encryption, imagine that you have to have a way to reverse that algorithm to recover the original password. A good hash algorithm resists this recovery, this is where encryption comes in. There is a conversion (like password to number), but there is also a reverse (like recover the password from the number). So for the one way hash, where the original sequence cannot be (should not be) recoverable, I have literally used hash. The food is a combination of ingredients is specific proportions that produce a unique flavor once combined. Without all sorts of chemical analysis (the weakness) the original recipe cannot be discovered. All the ingredients are put into a grinder which homogenizes them into a single consistent product. Unable to pick out the original pieces this is hash! Now for math purposes it can be more specific where original text is mutated into something else that is simpler to manage and does not require as much protection.

For example, if we do a simple hash algorithm which assigns all the letters of a password to a number and adds all those numbers together we get a number from which it is impossible to retrieve the original password (unless it was something simple or very repetitive). While that protect the original password, it has a weakness because many other passwords could produce the same number (collisions); “password” and “drowssap” would produce the same number as would “ssawprod” and many others.

So the true objective of the math for a one way has is to generate a unique value and reduce (or eliminate) the potential of collisions.

With encryption, imagine that you have to have a way to reverse that algorithm to recover the original password. A good hash algorithm resists this recovery, this is where encryption comes in. There is a conversion (like password to number), but there is also a reverse (like recover the password from the number).

]]> Comment on Weighing Coins during the Mystery Hunt by Philippe Fondanaiche http://blog.tanyakhovanova.com/?p=448#comment-311874 Philippe Fondanaiche Wed, 01 May 2013 13:35:16 +0000 http://blog.tanyakhovanova.com/?p=448#comment-311874 It is impossible to send the decision table.Last try: There are still two weighings and each one has three modes: balance (=), left pan heavier (>), left pan lighter ( < (2,3,4)/(5,6,8) = = > < = < > The table is read as follows: if the results of the 5th and 6th are weighed (), it is numbered 4 ... etc. It is impossible to send the decision table.Last try:

There are still two weighings and each one has three modes: balance (=), left pan heavier (>), left pan lighter ( <
(2,3,4)/(5,6,8) = = > < = < >
The table is read as follows: if the results of the 5th and 6th are weighed (), it is numbered 4 … etc.

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