Problematics | A boy and his kite

Bhaskara’s Lilavati, dedicated to the mathematician’s daughter and studied by scholars and puzzlers worldwide, often straddles the line between academic problem-solving and recreational puzzling. As we have noted before, the dividing line is a difficult one to define, and many of the problems in Lilavati underline the fact that it’s possible for a textbook-type exercise to be an entertaining puzzle at the same time.

One hurdle with bringing any of Bhaskara’s problems to Problematics, of course, is the fact that these problems are so widely circulating in the public domain that a Google search would lead to the solution immediately. As a way out, I have taken one of Bhaskara’s problems (from a secondary source that attributes it to Lilavati) and changed the numbers as well as the story it is situated in. The puzzle isn’t particularly difficult, either in Bhaskara’s original form or in my adaptation, but it should be fun to solve.

#Puzzle 93.1

A boy flying a kite fails to clear a coconut tree, with the result that the string gets caught in the branches. When the boy tries to yank the string free, he ends up snapping it. The kite drifts down and settles on a point on the tree that is at a height of 18m from the ground. The dejected boy rolls in his remaining string and watches the kite wistfully from a distance that is equal to three times the height of the kite. Then a gust of wind comes to the rescue, dislodging the kite from the tree and blowing it in the direction of the boy. The delighted boy runs towards the kite (which in effect means running towards the foot of the tree, given that the boy cannot fly like Peter Pan). The kite, meanwhile, travels along a line that is at an angle to the ground. Eventually, when the boy picks up the kite from the ground, the two have travelled an equal distance along their respective paths.

How far is the boy from the tree when he is reunited with his kite?

#Puzzle 93.2

1. The above cells, once filled with the appropriate letters, will spell out the title of a Hollywood classic with an ensemble cast

2. The title consists of two common words of 5 letters each

3. No letter is repeated

4. Letters 1, 2, 9, 8, 3 spell out the first name of one of the lead actresses in the film

5. Letters 1, 3, 2, X, 7 spell out her surname, where X is a letter that does not appear in the movie’s title

6. Letters 6, 7, 3, 2, 5 spell out a verb meaning to collect

7. Letters 10, 7, 5, 1, 9 spell out a verb meaning to accommodate

8. Letters 10, 9, 3, 2, 4 spell out a verb meaning to grasp

What is the film? (Hint: It is even older than 1939’s Gone With the Wind)

#Puzzle 92.1

Hi Kabir,

Two patterns among the numbers on each die are observed: (1) The tens digit on all 6 faces of a given die is the same; (2) The sum of the units digit and the hundreds digit on all 6 faces of a given die is the same.

These values are shown in the table. The sum of the tens digits across the five dice is D = 30. The sum of the units and hundreds digits across the five die is S = 47.

After rolling the five dice, the numbers showing up on the top faces are to be summed. The sum can be expressed as follows:

(100 x sum of hundreds digits) + (10 x sum of tens digits) + (sum of units digits)

= 100 (S – sum of units digits) + (10 x D) + (sum of units digits)

= 100 (47 – sum of units digits) + (10 x 30) + sum of units digits

= 100 (47 – sum of units digits) + (100 x 3) + sum of units digits

= 100 (50 – sum of units digits) + sum of units digits

So, the magician's trick is just to add up the units digits of numbers showing up on the top faces. Let this be N. Then the required answer is 100 (50 – N) + N. For the numbers given on the five dice, N would be in the range 5 to 39 and will form the two rightmost digits of the answer, and (50 – N) would be in the range 11 to 45 forming the two leftmost digits of the answer.

— Professor Anshul Kumar, Delhi

#Puzzle 92.2

Dear Kabir,

Man, wife, son, daughter, mother -> Their ages are in the form 18x,12x, 2x, x, 33x. So, (36, 24, 4, 2, 66) seems more realistic than the other possible but not-so-realistic set (54, 36, 6, 3, 99).

— Sampath Kumar V, Coimbatore

Some readers have solved #Puzzle 92.1 only partially, mostly observing how to work out the sum but not completely explaining why this method works. That is why there are two separate lists for those who have solved it.

Solved both puzzles: Professor Anshul Kumar (Delhi), Sampath Kumar V (Coimbatore), YK Munjal (Delhi), Akshay Bakhai (Mumbai), Shishir Gupta (Indore), Sundarraj C (Bengaluru), Anil Khanna (Ghaziabad)

Solved both puzzles (partially): Yadvendra Somra (Sonipat), Sunita Gupta (Delhi), Raghunathan Ravindran (Coimbatore)

Solved #Puzzle 92.2: Hiten Jindal (Delhi), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), Ajay Ashok (Mumbai)

Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com