The agony and ecstasy governing problems using words or, more accurately, where a bunch of them are given and you have to then figure out what connects them or delve a deeper relationship is that I usually used to give 26 of them. My bad. Because you instantly realised there were 26 alphabets in the E language and immediately got to work. So I’ve decided on a sneakier approach: give less words (but you still might not know if they were all actually 26 or whittled down).
Right, so what’s the connection between: WISDOM, BLESSING, UNKINDNESS, KNOT, HOVER, DESCENT, SCOLD, BLOAT, BATTERY, OSTENTATION?
THROUGHPUT
(The stuff from the dark ages was: “Why is it impossible to make a model of an irregular polyhedron that’s unstable on every face?”)
You can’t have an irregular polyhedron with all unstable faces because whichever face you place it on, it would topple to the next face, which is also unstable making it topple again, and so on indefinitely. That would create a perpetual motion machine, which we all know is an impossibility! -- J R K Rao, jrkrao@gmail.com
(And here are two pending alt answers.)
Regarding how far a dog can run into a forest, what if it did not run straight? What if it was ‘in the woods’ and ran zigzag or in circles there? Also observe the dog from say 3/4th of the way. From here, it would still be running into the woods, even after crossing the halfway mark. That is the theory of dog relativity! -- Dr Ramakrishna Easwaran, drrke12@gmail.com
My answer to the counter-question posed by K T Rajagopalan is the smallest number consisting of only 7’s and divisible by 199 is an 11-digit one, which is 2 followed by ten 7’s -- i.e, 27777777777. When divided by 199, it yields a quotient of 139586823. -- K Narayana Murty, k_n_murty@yahoo.com
(The second problem was: “The two numbers 220 and 284 have a special connection. What is it?”)
220 and 284 are amicable numbers; that is, the sum of the factors of 220 and 284 is equal to the sum of the proper divisors of the other number. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220 and the factors of 284 are 1, 2, 4, 71, 142, 284. After removing the integers, 220 and 284 from both the lists and calculating the total of the remaining factors for each number, we get 284 and 220. -- Rekha G, g.rekhapai@gmail.com
The famous mathematician S Ramanujan explained that an ideal friendship should be like these numbers to complement each other -- that even when one is absent the other should represent the other! -- Ajit Athle, ajitathle@gmail.com(Among the first billion who also got it right are exactly those people who know who they are. Meaning simply add your monikers in your heads and we’ll get them telepathically. Also, hey problem submitters how about submitting some tougher ones in future so I don’t have to go through half a mil emails? -- MS)
(The third one was: “In images of Mt Fujiyama is there any way to make out the difference between the volcano and its reflection?”)
The only method is that the left side and the right side of the image will coincide with the left and right sides of the volcano respectively. The base of the image also coincides with the base of the mountain. (Not really -- MS) -- Shashi Shekher Thakur, shashishekher@yahoo.com
Since the camera is never held at the water level but about five to six feet higher at eye level, tell tale differences can be made out on closer examination of the actual image and its reflection. -- Dhruv Narayan, dhruv510@gmail.com
BUT GOOGLE THIS NOW
1. There’s a hot cup of coffee and room-temperature cream. You want to wait a few minutes before you drink the coffee, but you want it to be as hot as possible. Should you pour the cream in the coffee immediately, just before you drink it, or does it matter?
2. Suppose we set a small circle rolling around the interior of a large circle of twice its diameter. Now follow a point on the small circle. What English inventor exploited this principle in the 19th century to produce a hypocycloidal engine in which a steam piston drives a wheel?