Although it is not really necessary, I often check if the numbers I bring to the fictional setting of my puzzles are realistic or not. For example, what is the typical range of speeds in a motorcycle race? Had I made the participants too fast? So I checked and was astounded to find that speeds can exceed 350 kph in the premier class.Problematics (Unsplash)The speeds you will determine in the puzzle below are far below that range. We can safely assume that the participants are amateur motorcyclists. That said, the speeds are still far above what one would call safe in regular driving. So here’s your next speed-and-distance puzzle.#Puzzle 197.1A motorcycle race is taking place on a long, empty highway in America. With that country still following the imperial system of measurement, speedometers on motorcycles and other vehicles are marked in miles per hour instead of the metric km per hour, so we will use that unit.The race starts at 9 am. It covers the entire highway.One of the motorcyclists is a considerably faster than the rest. The second and third fastest riders are closer to each other, a difference of 3 mph, which enables the second fastest to finish 3 minutes earlier than the third fastest. In contrast, the winner is 15 mph faster than the second fastest, and finishes 15 minutes before the third fastest.What is the length of the highway, what are the speeds of the top three, and at what times do they finish their races?#Puzzle 197.2Two friends from school, both now grown up, have a chance meeting at a mall. They ask about each other’s jobs and families and find that one is a dentist and the other a businesswoman. Each woman has two children: an elder daughter and a younger son. The dentist’s daughter is older than the businesswoman’s daughter, who is older than both boys.The ages of the four children, in fact, are integers in arithmetic progression. The cube of the age of the dentist’s daughter is equal to the sum of the cubes of the other three children. All four are minors.How old are the four children?MAILBOX: LAST WEEK’S SOLVERS#Puzzle 196.1Let the three gangsters be A, B, C and their respective sons be a, b, c.The conditions are: a boy can stay with men only if his own father is among them; and boys alone are always safe.A valid sequence of crossings is shown in the table.— Vinod Mahajan, New Delhi#Puzzle 196.2Hi Kabir,The total time taken is 5 hours.X and Y travel on X's scooter for 36 km at 20 km/h. This takes 1.8 hours. X drops Y, turns back and travels for 24 km, taking 1.2 hours. During this time of 1.8 + 1.2 = 3 hours, Z walks 12 km at 4 kph, which is 24 km from the point where X dropped Y.X now picks up Z and they travel the remaining distance (24 km + 16 km to final destination), which takes 2 hours by scooter. Meanwhile, Y walks 16 km at 5 km/h for 3.2 hours. The total time taken by all three is 5 hours.This can be determined as follows. Let ‘a’ be the distance travelled together by X and Y. During this time, Z, walking at 1/5 of the scooter speed, travels a/5. So when X drops Y, they are ahead of Z by a distance of a – a/5 = 4a/5. When X turns back to meet Z, the scooter travels 5/6 of this distance and the walker travels 1/6, i.e. X travels for 5/6 x 4a/5 = 4a/6 before meeting Z. After picking him up, the scooter travels the same distance back to the point where Y was dropped, and then the remaining (52 – a) km.Total distance travelled by X = a + (4a/6) + (4a/6) + (52 – a)Total time taken by X = 1/20 [a + (2)(4a/6) + (52 – a)]Total time taken by Y = (a/20) + (52 – a)/5Equating the above two times gives a = 36 km.— Sabornee Jana, Mumbai#Puzzle 196.2Hi Kabir,There are two possibilities:(i) X carries Z on his scooter for some distance, goes back to pick up Y and carries him to the destination.(ii) X carries Y on his scooter for some distance, goes back to pick up Z and carries him to the destination.Whichever order is followed, the total time taken turns out to be the same. It is 5 hours.If Z is carried on the scooter first, they travel for 2 hours (40 km), after which X drops Z and goes back to pick up Y and take him to the destination.If Y is carried on the scooter first, they travel for 1 hour 48 minutes before X drops Y and goes back to pick up Z and take him to the destination.— Professor Anshul Kumar, New DelhiSolved both puzzles: Vinod Mahajan (Delhi), Sabornee Jana (Mumbai), Professor Anshul Kumar (Delhi), Shri Ram Aggarwal (Delhi)Solved #Puzzle 196.1: Yadvendra Somra (Sonipat)Solved #Puzzle 196.2: Dr Sunita Gupta (Delhi), Ajay Ashok (Delhi)Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com