When an engine is hauling a train from the front it does so by pulling on the couplings between each of the coaches behind it. You know that of course, right? As a result the two bumpers that each of those coaches also have in the back get slightly distanced from the two bumpers the coach behind it has in its front. However, if an engine is instead pushing a train from the back, the couplings are no longer needed because the bumpers do the needful. Meaning all of them are in constant touch, pushing each coach in front.
Now comes the fun part. Sometimes in hilly areas two engines are required to make the rake move against an uphill gradient with one in front and one in the back so that the net effect is that the train is being pulled and pushed at the same time. Still right, right? So what happens to the bumpers. Are they touching or separated?
THROUGHPUT
(The old one was something about a celebration on Bell island which is way too long to go into without all of us going to sleep. So the answers have to fill in the blanks.)
Number of minutes in a day = 1440. The bells ring alternately implying that we are dealing with x and (x + 1) minutes of time intervals. Further, x*(x + 1) = 1440*N where N is prime. The easiest way this could happen is if 1440 is one of the two intervals and the other being either 1439 or 1441. Of these two, 1439 is a prime number. Or N = 1439 or the time intervals are 1439 minutes and 1440 minutes and thus the two bells chimed together at 12 noon 1439 days ago. -- Ajit Athle, ajitathle@gmail.com
Also, the LCM of the ding-minutes and dong-minutes is 1440*P, where P is a prime factor of one and only one of the two durations. I noticed that 1439 is a prime, giving the bells ding-dong at 1d and (1d - 1m), or equivalently, 1440m and 1439m. -- Dhruv Narayan, dhruv510@gmail.com
(The second problem was: “If there was a litre more of wine and a litre less of water in a mixture of the two the ratio would have been 7:8. But if there was a litre more of water and a litre less of wine the ratio would have been 2:3. What’s the original mixture?”)
The total volume of the original mixture is 30 litres; wine 13 litres and water 17 litres, the ratio being 13:17. When the wine is one litre more and the water is one litre less the ratio becomes 14:16, or 7:8. When the wine is one litre less and the water is one litre more the ratio becomes 12:18, or 2:3. -- Dr P Gnanaseharan, gnanam.chithrabanu@gmail.com
Original mixture is of 30 litres. So simple that it is unqualified to be included in your column. -- Seshagiri Row Karry, srkarry@yahoo.com.
(Therefore among the first million to get it right are: Vaishnavi Racherla, vaishnavi2000@gmail.com; Ganesh Ram Palanisamy, 1969ganram@gmail.com; K Balaji, mayavarathan29@rediffmail.com; Shashi Shekher Thakur, shashishekher@yahoo.com; J Vaseekhar Manuel, orcontactme@gmail.com. -- MS)
(The third one was: “What do Bedecked, Icebox, Choice and Kidded have in common?”)
Written in caps, a straight line passing through the middle cuts the words in two mirror images. If the upper half is kept on a mirror, the complete word reappears again. Incidentally, to make the problem more difficult you should have changed the last word to “skidded” and then asked which word did not belong to the rest. -- Narayani Menon, narameme@gmail.com (Hey NM, how about letting me know things like this before I write them? -- MS)
The words in capital letters exhibit 180-degree horizontal symmetry. The capital letters A, M, T, U, V, W and Y are vertically symmetrical; the capital letters B, C, D, E and K are horizontally symmetrical; the capital letters H, I and X are both horizontally and vertically symmetrical and the letter O is infinitely symmetrical. -- Saishankar Swaminathan, saishankar482@gmail.com
BUT GOOGLE THIS NOW
1. Consider a four by four square grid. Name the top left square A and top right square B. In going from A to B what’s the maximum number of squares that a chess knight could touch, including A and B, if the knight makes only permissible moves, does not touch any square more than once and does not go outside the grid?
2. Which is the only Olympic Games which were not held in a leap year?
— Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)