Problematics | The perfect capital M

Those of us who have watched a child grow up, especially those who are parents, may have different memories of how the child first learned to write. In my experience, they begin with horizontal and vertical lines, first with a capital I and then going on to capital T, H etc. Later, once they learn how to draw circles and curves, they move on to capital O, capital P, etc. Somewhere along the way, they learn how to draw slanting lines, which gives them letters such as a capital M.

The capital M is the subject of the following puzzle. You can choose to keep it simple or make it complicated, but it requires nothing more than elementary geometry and algebra.

#Puzzle 96.1

A mathematician is teaching his young daughter how to write a capital M. He draws a horizontal rule across a blank sheet of paper and tells her: “Draw a vertical line on top of the rule here… Good, now draw another vertical there.”

The child’s two lines are perfectly perpendicular to the rule, the father observes with satisfaction. Their lengths, however, are different. I should have drawn another horizontal rule above the existing one, Dad tells himself in hindsight. That would have restricted both her verticals within these limits.

He desists from giving voice to his thoughts. Rather than discourage the child, he tells her: “Well done. Now draw a slanting line from the top of the left vertical to the bottom of the right vertical.” The child now protests. “But a capital M does not look like that. The slanting lines don’t extend to the bottom,” she says. The father sets her mind to rest: “Never mind, we will erase the unnecessary part later. For now, this will give you a sense of direction.”

When the child has drawn the slanting line, the father asks her to draw a second such line, this time from the top of the right vertical to the bottom of the left one. “We will delete the extra part here, too, won’t we?” says the child. “Erase, not delete,” Dad corrects her: “you should spend less time on my phone and more on your notebooks.”

When the two slanting lines are drawn, the mathematician notes that each perfectly touches the upper tip of one vertical and the bottom of another. He lets the child erase some of the extraneous portion, before completing the rest of the cleanup himself.

Being a mathematician, he cannot resist measuring the two vertical lines. One is 2cm and the other 3cm, give or take a few mm in both. He does not mention this, of course. “A perfect M,” he congratulates his daughter.

At what distance from the horizontal rule do the two slanting lines meet?

#Puzzle 96.2

Last week, you helped a traveller choose between two barbers, one with a neat haircut and the other with unruly hair. The people of the village, of course, already know who is the better barber. His business ebbing, the less efficient barber tries a trick to draw in more customers. He puts the following sign outside his shop:

WHAT DO YOU THINK

I WILL GIVE YOU A HAIRCUT

FOR FREE

A villager decides to visit the barber; although he knows the haircut will be poor, he is happy he won’t have to pay. After the haircut, to his surprise, the barber demands ₹50. The customer protests: “But your sign says, ‘What do you think! I will give you a haircut for free’. Why are you demanding payment now?”

“No sir,” the barber corrects him, “you punctuated the sign wrong.”

What alternative punctuation would imply that a haircut is not free?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 95.1

Hi Sir,

This is a simple problem with the application of geometric progression. If the deal between the conman and the neighbour was for 31 days, the conman would have received (2³¹ – 1) = 214,74,83,647 paise or ₹2,14,74,836.47. At the same time would have paid his neighbour 31 x 5,00,000 = ₹1,55,00,000. Thus the conman would have gained ₹59,74,836.47.

But when the neighbour offers a deal for 28 days, the conman would have earned (2²⁸ – 1) = 26,84,35,455 paise = ₹26,84,354.55 while paying the neighbour 28 x 5,00,000 = ₹1,40,00,000. In this case, the conman would have lost ₹1,13,15,645.45.

So the conman should NOT accept the new deal.

I found that the deal is in favour of the neighbour for the first 30 days and in the favour of the conman after 31 days or more.

— Harshit Arora, IIT Delhi

Harshit Arora has plotted a graph too, but the one being published here has been sent by C Sundarraj of Bengaluru. Sundarraj’s graph shows a day-by-day comparison between the cumulative deposits/earnings of the conman and the neighbour. Indeed, as many readers have observed, the advantage shifts from the conman to the neighbour on the 31st day.

#Puzzle 95.2

Hi Kabir,

As there are only two barbers in the village, they must be cutting each other’s hair. So, the haircut of the barber with a neat hairstyle was done by the barber with unruly hair. So our man should visit the barber with unruly hair.

— Shishir Gupta, Indore

Solved both puzzles: Harshit Arora (IIT Delhi), Sundarraj C (Bengaluru), Shishir Gupta (Indore), Akshay Bakhai (Mumbai), Shruti M Sethi (Ludhiana), Dr Sunita Gupta (Delhi), Sanjay S (Coimbatore), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat), Raghunathan Ravindranathan (Coimbatore), YK Munjal (Delhi), Ajay Ashok (Mumbai), Sampath Kumar V (Coimbatore).

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