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Problematics | Rescue dog

Nov 18, 2024 11:43 AM IST

You know the length of the ladder being used to rescue a dog from a window. Can you work out the height of the window?

In the more than two years that Problematics has been running in HT, we have managed to set ourselves a template for the difficulty level of our puzzles. The puzzles are intended to challenge readers and still be gettable in the end. That said, there will be puzzles from time to time that will be more difficult than usual. Last week’s clock puzzle was such an exception, as it appears to have stumped several readers.

Representational image.(Pixabay)
Representational image.(Pixabay)

To compensate, this week’s puzzles are deliberately being kept easier than usual. They do not, however, deviate from the other key aspect of our template: you should find them fun to solve.

#Puzzle 117.1

Puzzle 117.1
Puzzle 117.1

A fireman sets out to rescue a dog trapped in a high room using a ladder. On the ground, a mathematician tries to figure out the height of the room’s window, where the dog is waiting expectantly to be rescued. The fireman is not in the least bothered by these trivialities; all he knows is that his ladder is long enough to reach the window.

“How long is the ladder?” the mathematician calls out. “Exactly 4m,” the fireman replies, as he rests the top of the ladder right against the window.

Realising that this is all the information he will get out of the fireman, the mathematician observes the surroundings for additional data. He notes that the ladder just touches the corner of a square slab that rests against the wall. He brings out his measuring tape and finds that the slab is 1m to a side.

How high is the window from the ground?

#Puzzle 117.2

19 x 19 = 519

In the equation above, every digit represents a different digit. Needless to say, all the 1s represent the same digit, and the 9s represent another digit that is the same for every 9.

If the multiplication is correct, what does each digit represent?

MAILBOX: LAST WEEK’S SOLVERS

Puzzle #116.1

Dear Kabir,

If the sum announced is 12 or less, half of it will be the number on the die. If the sum announced is more than 12, then subtract 12 and halve the difference to get the number on the die.

If number thought of is (12n + m) where n= 0, 1, 2 and so on, and m is 1, 2... 12, and if the number on the die is x (which may 1 to 6), the sum will turn out to be either be (2x) or (12 + 2x). In the former case it will be (2, 4, 6... 12) and in latter case it will be (14, 16, 18... 24).

— Yadvendra Somra, Sonipat

#Puzzle 116.2

Puzzle 116.2
Puzzle 116.2

Hi Kabir,

Let H and M denote the angles made by the hour and minute hands, respectively, with respect to 12 o'clock position, when the time is h hours and m minutes. Then M = 6m degrees and H = 30(h + m/60) degrees. That is,

H = 30h + M/12

If the two hands are interchanged, then by symmetry,

M = 30h' + H/12, for some h' (0 < h' < 12)

These two simultaneous equations can be used to solve for H and M in terms of h and h'. We are not interested in actual solutions, but only the count of distinct solutions. Since there are 12 possible values each of h and h', there are 144 solutions. However, it may be noted that the solution corresponding to h = h' = 11, is H = M = 360°. This is the same as H = M = 0 degrees, which corresponds to h = h' = 0. Therefore, there are 143 distinct solutions.

The graph helps in visualising the solutions. Green lines represent the first equation for 12 different values of h and the red lines represent the second equation for 12 different values of h'. These two sets of lines intersect in 144 points corresponding to 144 solutions. The top-rightmost point and the bottom-leftmost points represent the same solution.

— Professor Anshul Kumar, Delhi

In hindsight, the clock puzzle was so tough that it could have been the first of last week’s two puzzles. Apart from Professor Anshul Kumar, only Dr Sunita Gupta got it correct. Some other readers either gave a handful of examples, or misunderstood the question itself.

Solved both puzzles: Professor Anshul Kumar (Delhi), Dr Sunita Gupta (Delhi)

Solved #Puzzle 116.1: Yadvendra Somra (Sonipat), Shri Ram Aggarwal (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Sabornee Jana (Mumbai)

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

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