Problematics | Matches with sachets

There have been some occasions when I have simply lifted a puzzle from an existing collection and presented a modified version with due credit. With a festival due on the date of publication of this episode of Problematics, this is one of those occasions. I need to rush this, so I am depending on the readymade writings of a legend, Martin Gardner, who does not specify whether he created the puzzle or sourced it from somewhere else.

Because Gardner’s work is widely available on the internet, I have changed the packaging of his original. It is one of those puzzles that appear to have incomplete information, but which become solvable once you get down to them with the limited data that is given.

While still on incomplete information, we are also repeating last week’s record player puzzle (#135.1) this week. That is because different readers have approached it differently, so I have clarified what the puzzle really asks of its solvers. Also, readers have observed that one piece of data is absent (although that value could have been assumed) and I have now added that variable in the rerun. But before that, here are your fresh puzzles for the week.

# Puzzle 136.1

A shopkeeper orders a number of matchboxes and an equal number of shampoo sachets. He pays ₹2 for each sachet and ₹1 for each matchbox.

The shopkeeper sells every item at a 10% profit, and the time soon comes when he has recovered his original investment. He checks his stock and finds that he has a total of 7 items still unsold. To prevent any misunderstanding, one item can be either a matchbox or a shampoo sachet.

These 7 last items, once sold, will bring him an amount that will be his profit. How much is that profit?

# Puzzle 136.1

In an apartment building, a number of students have sat for their HS board exams. When the results come, the apartment is soon filled with gossip about 9 students who have failed in mathematics and 11 who have flunked English. Among them, the number of students who have failed both subjects is three times the number who have passed both.

How many students from the building have taken the exam?

MALBOX: LAST WEEK’S SOLVERS

#Puzzle 135.1

Hello Kabir,

To the best of my understanding there can be three answers to this question. A definitive answer can only be provided for the first option, given the limited information available.

(1) The stylus moves 90mm if we consider its movement in a straight line. For a disc of radius 150mm, 10mm and 50mm are taken up by the lead-in groove and the inner unrecorded circle respectively.

(2) Since we do not know the length of the tonearm...*

(3) Some may say the distance travelled by the stylus could well be the entire length of the music groove. Although the needle is not actually moving in that direction, it could be construed as having moved in the opposite direction of the movement of the disc — much like on a treadmill. Had we been given the number of revolutions the disc takes to play out the music piece, we could have calculated the distance “travelled”.

I very vividly remember the manual gramophone that my maternal grandfather had in Chandigarh in the 1960s (exactly as in the HMV logo). Although the instrument has been discarded, I have dozens of 78 RPM vinyl records - but no equipment to play them on.

— Sanjay Gupta, Delhi

* I have edited out part of Sanjay Gupta’s mail to avoid giving the game away. I was not looking for either Option (1) or Option (3), but for Option (2). In fact, different readers have used different approaches from among the three highlighted above.

Yes, I missed giving the length of the tonearm, but then I expected readers would make assumptions (as Shishir Gupta of Indore did). Here is the rerun of the puzzle.

An LP disc of 300mm diameter has a lead-in groove 10mm wide, and an inner unrecorded circle of diameter 100mm. The tonearm is 250mm long from the pivot to the needle or stylus. When the record plays, ignore the grooves. We simply observe the needle’s movement in terms of the path it traces relative to any fixed point in space.

How far does the stylus travel if the full record is played?

#Puzzle 135.2

Hi Kabir,

The original word is NOWHERE. And split by a space bar, it becomes NOW HERE.

— Shishir Gupta, Indore

In the credits rolling below, I have acknowledged every reader who has attempted the first puzzle, although the replies do not match one other. Now that I have provided the missing piece of data, I am sure everyone will get the same result this time.

Solved both puzzles: Sanjay Gupta (Delhi), Shishir Gupta (Indore), Vinod Mahajan (Delhi), Sampath Kumar V (Coimbatore), Dr Sunita Gupta (Delhi), Dr Vivek Jain (Baroda), Yadvendra Somra (Sonipat), Shri Ram Aggarwal (Delhi), Ajay Ashok (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Professor Anshul Kumar (Delhi), YK Munjal (Delhi), Aishwarya Rajarathinam (Coimbatore)

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