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Problematics | Trains meeting trains
This week, a puzzle with four intertown trains involving speed and distance, and a crossword without clues for the states of the Union.
The number of different forms that speed and distance puzzles can take appears to be countless. I thought I had seen them all when I came across a new form in a collection of puzzles from Henry Ernest Dudeney put together by Martin Gardner. Below are two of Dudeney’s puzzles (with very slight modifications), but one of them belongs to a variety that has appeared in Problematics earlier.
#Puzzle 184.1
There are two towns, X and Y, between which four trains every day. That is to say, two of them run from X to Y, and two from Y to X. All four trains leave their respective stations at exactly the same moment but their arrival times are not all the same. Each train, of course, has its own constant speed.
As an experiment, two puzzlers journey separately on the two trains from X to Y. One puzzler meets opposite train A at a point 120 km from X. After travelling another 20 km, he meets opposite train B.
The puzzler on the other XY train meets opposite train A after travelling 126 km from Y. When she meets opposite train B, she is exactly halfway between the two towns.
The above might be a bit easy for Problematics readers, so here is a bonus puzzle. There are two more towns, P and Q, and a traveller takes a train from P to Q, a journey that is partly on level terrain and partly uphill. The train’s uphill speed reduces to 60% of its level-ground speed.
As of now, the transition from flat to uphill happens exactly 1 hour after leaving P. Had the entire route been flat, the train would have reached Q exactly 2 hours before its actual arrival time. Had the uphill terrain had started 50 km farther from where it actually began, the total journey would have been 80 minutes shorter.
What are the distances between X and Y, and P and Q?
#Puzzle 184.2
The above crossword grid can be filled with the names of all 28 full states of the Indian Union, plus Delhi. No Union Territories other than Delhi are included. The size of the grid (28 x 18) was the tightest I could manage, with all the entries predetermined.
(Spellings used: ANDHRA PRADESH, ARUNACHAL PRADESH, ASSAM, BIHAR, CHHATTISGARH, DELHI, GOA, GUJARAT, HARYANA, HIMACHAL PRADESH, JHARKHAND, KARNATAKA, KERALA, MADHYA PRADESH, MAHARASHTRA, MANIPUR, MEGHALAYA, MIZORAM, NAGALAND, ODISHA, PUNJAB, RAJASTHAN, SIKKIM, TAMIL NADU, TELANGANA, TRIPURA, UTTARAKHAND, UTTAR PRADESH, WEST BENGAL)
Can you fill up the grid with the right states in the right slots? It is much easier than it looks.
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 183.1
Dear Kabir,
The error in Mr Small’s argument comes after the step
(s – n)² = (b – n)²
Only the positive square roots have been considered, i.e. (s – n) = (b – n). But it should be
±(s – n) = ± (b – n)
If we take (s – n) = –(b – n), we get s + b = 2n, which was the assumption of Mr Small in the beginning.
— Yadvendra Somra, Sonipat
#Puzzle 183.2
Dear Kabir,
The probability of a boy sitting on the first end = 3/7
The probability of a second boy sitting on the other end = 2/6
Total probability of two boys sitting on the ends = (3/7)*(2/6) = 1/7
— Dr Sunita Gupta, New Delhi
Solved both puzzles: Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Vinod Mahajan (Delhi), Ajay Ashok (Delhi), Professor Anshul Kumar (Delhi)
Solved #Puzzle 183.2: Dr Nitin Trasi (Sydney)
