Interesting paradoxes arise from attempts to classify numbers as ‘interesting’ or ‘dull’.
For example, 1729 is often considered an interesting number, because it is the smallest number expressible as the sum of two positive perfect cubes in two different ways. (The number is also famous in mathematics lore, for it was Srinivas Ramanujan who once pointed out its interesting property to G.H. Hardy). However, attempting to classify all numbers this way leads to a paradox (strictly speaking, an antinomy). In a classification of numbers as to whether they have interesting properties or not, there would be a smallest number with no interesting properties whatsoever. For instance, all the numbers up to 37 may have interesting properties, but 38 may not.
This in itself would be an interesting property of the number 38, so it would no longer be dull. This is a classic paradox of self-reference, or proof by contradiction.
PUZZLE 1: Spike has a large bag of candies, each of which is one of 5 possible different flavours: apple, banana, cherry, Dutch chocolate, and elderberry. Assuming he can fit up to 10 candies in his mouth at once, how many different flavours can he make' Note that one apple and one banana is the same flavour as two apples and two bananas (just a larger amount), but that one apple and two bananas are not the same as two apples and one banana. Also note that Spike has more than 10 candies of each flavour.
PUZZLE 2: N teams compete in a tournament. Each team plays every other team once. In each game a team gets 2 points for a win, 1 for a draw and 0 for a loss. Given any subset S of teams, one can find a team (possibly in S) whose total score in the games with teams in S was odd. Is N is an even number'
Solutions on November 7
Manas Das, Cal-47; Rishab Jain, Cal-26; Devanu Ghosh Roy, Cal-94; Debanshu Sinha and Soubhik Sarkar, Jamshedpur; A.K. Majumdar, Cal-106; Shayak Bhattacharjee; Abhijit Chatterjee, Cal-38; Debratna Nag, Jadavpur; Sayan Chatterjee; Soumava Chakraborty; Samudra Banerjee; Piyush Modi; Biswa Ranjan Bhattacharya, Cal-41; A.M. Ghatak; Sayan Chatterjee; Howrah Zilla School; Sugato Lahiri, Guwahati; Arun Neogi; Edwin King, Alameda-California; Toofan Majumdar, Howrah-3; Sreechandra Banerjee; Subhabrata De, Cognizant Technology Solutions; Praveen Dhanuka, Jadavpur University; Shayak Bhattacharjee; Pratiti Mandal, Cal-34; Soumya Ranjan Khillar, NIT-Rourkela; Sugata Sadhukhan, Rasulpur; Amitabh Roy Sharma, Bokaro Steel City; Mukulika Jana, Chikrand, Hooghly; Sudipta Chakraborty, TCS-Cal; Debasis Bhattacharyya, Kalyani Govt. Engg. College; Madhumita Pal, Techno India-Salt Lake; Debasis Ganguly, Alumnus Software; Sundaresh Shrikant, Jamshedpur; Manisha Mukherjee, Madhyamgram; Ravi Raja, Cal-20; Arka Sarkar, Jadavpur University; R. Ranjit, Tamluk; Sanjeev Kedia, Bangur.
Solution 1: The back man can see the hats worn by the two men in front of him. So, if both of those hats were white, he would know that the hat he wore was black. Since he doesn’t answer, he must see at least one black hat ahead of him. After it becomes apparent to the middle man that the back man can’t figure out what he’s wearing, he knows that there is at least one black hat worn by himself and the front man. Knowing this, if the middle man saw a white hat in front of him, he’d know that his own hat was black, and could answer the question correctly. Since he too doesn’t answer, he must have seen a black hat on the head of the front man. After it becomes apparent to the front man that neither of the men behind him can answer the question, he realises the middle man saw a black hat in front of him. So he says, correctly, “My hat is black.”