Okay here’s the thing. Now that computers have beaten the best human grandmasters in chess, presumably they would be able to solve the toughest chess problems too because of brute number crunching power and tactics. But watch this deal: could they also set chess problems that no human grandmasters can solve? They should, right?
Three chess players agree to play a series of games for a prize which would go to the first player to win two consecutive games. The players draw lots to see which two shall play the first game and from then on the winner plays against the person who just sat out. Assuming the three players to have exactly equal skills, what are their respective chances?
THROUGHPUT
(“There are 100 marbles in a bag numbered 0 - 99. You draw them out one at a time, noting the number and replacing it. On average, how many numbered 1 - 99 will you have to draw one or more times before getting to marble number 40?”)
It is clear that marble 0 should not be drawn before number 40 is drawn. Among 0 and 40, the probability of drawing 40 earlier than 0 is 1/2. The same holds good for every marble 1 to 99. The probability of drawing each of them at least once before drawing 0 is 1/2. The average number of draws will be a statistical expectation which is the total number multiplied by probability. It becomes 99*1/2 = 99/2. The same question can be asked for any marble from 1 to 99. Arithmetically --Abhay Prakash, abhayprakash@hotmail.com
(The second problem was: “You’re rowing a boat on a small lake and throw the anchor overboard. Does the water level of the lake rise, fall or stay the same?”)
While the anchor is in the boat, its weight displaces a volume of water V1 equal in weight to the anchor (by Archimedes’ principle). After being thrown overboard, the sunken anchor displaces a lesser volume of water V2, equal in volume to the anchor. V2 is less than V1 because the anchor is denser than water (otherwise it would not sink!) So, with less water displaced, the level of the lake falls. -- Dhruv Narayan, dhruv510@gmail.com
By throwing the anchor off the boat overboard it makes the boat’s weight to come down and the displaced water volume of the lake to also come down by changing the water level marginally. -- Nrusingha Behera, ncb123.age@gmail.com
Considering that the anchor is made of wrought iron, it would be denser than water. The displacement of the anchor as mass when on the boat is greater than its displacement as volume when in water. Thus, if you throw it overboard the water level will fall. -- Shashi Shekher Thakur, shashishekher@yahoo.com
(The third one was something that people actually fell for! I had said, “Can a bat commit suicide by tying its neck to the floor and having its legs suddenly attached to a large helium balloon?” I had only asked for serious answers. People took that seriously too.)
A typical helium filled balloon on a ribbon can lift approximately 4.8g. If the balloon is thrice as large as a normal one, it can still lift only 14.4g. But a bat’s minimum weight can be approximately 270g, so the balloon does not have sufficient force to pull the bat upwards, making the bat just lie on the ground. -- Lipika Muthu, geelipm@gmail.com
Suicides are peculiar to humans only. So your question should be read as “Can a bat kill itself . . . ” and so on. The bat, however, gets killed due to the tearing of the body into two parts, one above its legs and the other flying with the balloon. -- Seshagiri Row Karry, srkarry@yahoo.com
BUT GOOGLE THIS NOW
1. Find the smallest word such that taking some of the letters from it, you can get four different words from the following list: ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE (yes, there’s no TEN). The word should contain no other letters except those mentioned in the list. (For instance, if you want to make SEVEN or THREE, you need to have two E’s in the word.) You can also try to find similar words, one letter longer.
2. Find the smallest word such that taking some of the letters from it, you can get six different words which are the letter representations of the numerals 1 to 20. The word should contain no other letter except those mentioned in the list.
Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)