# Analytical Engines

### By Mukul Sharma

Published: Thu 5 Jan 2017, 11:00 PM

Last updated: Fri 6 Jan 2017, 1:00 AM

E4
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. 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. So what happens to the bumpers. Are they touching or have they separated?

DEAR MS
(The old one was about a celebration on Bell island, which is way too long to go into without all of us falling asleep. So the answer will have to fill in the blanks. - MS)

Alexander-Graham-Dept:
Since the bells ring alternately and also a minute apart, let us say that they ring at x and (x+1) minutes. Knowing that there are 24 x 60 = 1440 min in a day; we can write x*(x+1) = 1440*N, where N is a prime number. The obvious choice, as seen in the equation, can be 1439 or 1441. Of these two, 1439 is prime. Thus, the time intervals are 1439 min and 1440 min; and so the two bells coincided at noon 1439 days ago.
- Saifuddin S F Khomosi, saif_sfk@hotmail.com

(The second problem was: "If there was a litre more of vinegar 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 vinegar, the ratio would have been 2:3. What's the original mixture?" - MS)

What-A-Ratio-Dept:
The original mixture has 13 parts vinegar and 17 parts water. With the given data, I created two equations using v for vinegar and w for water and then solved for v and w. The solution was too sour for my taste, but hopefully good enough to share!
- Ramesh S Mahalingam, ramesh@idealmc.com

Initially, there was 13 litres of vinegar and 17 litres of water. It was a simple problem of ratios. Thank you for giving a simple problem like this (I am only a student and never get to solve your riddles because they are beyond my knowledge and scope).
- Joel Chacko Jacob,
joelcjacob@yahoo.co.in

Your page is always full of tricky problems. This time, it was a surprisingly simple problem based on a two variable algebra sum! Call vinegar v and water w. You then get two equations with two unknowns. When solved, you have v = 13 litres and w = 17 litres. Crosscheck tallies well with 14/16 or 7/8 and 12/18 that is 2/3! Incidentally, which was the site that sends emails that don't open?
- Benji Barker,
aratimu@yahoo.co.uk

(The third one was: "What do Bedecked, Icebox, Choice and Kidded have in common?" - MS)

Mirror-Images-Dept:
The words BEDECKED, ICEBOX, CHOICE and KIDDED in caps all have horizontal symmetry. Draw an imaginary horizontal line through the words and you find that both the sides are the same.
- Jaelyne Tauro, jaelynetauro@gmail.com

ENDGAME(S)
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. Why were the Olympic Games not held on a leap year only once?

(Mukul can be reached at mukul.mindsport@gmail.com)

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