Problematics | An accountant’s missing figures

First there were the single-screen cinemas, the only place where we could watch films. Then the 1980s brought cinema to our home via videocassettes, which gave way to DVDs some two decades later. In between, there was a brief phase when we watched films on video CDs, but this was something largely limited to Asian countries. Eventually, DVDs too gave way to digital video files, and collectors would store these files on external hard drives when their computers ran out of space.

Today, of course, the most popular mode of watching films at home is on OTT. That does not mean digital files have gone out of fashion, for some OTT platforms give subscribers and/or buyers the option of downloading the movies. As a longtime collector, I still have my old DVDs (I never had my own VHS player) but store all my new movies on external hard drives.

The following puzzle is based on such hard drives. To be honest, it would have worked with any other item that could be bought at the same price, but I felt it would become more fun if I set it around something related to movies.

#Puzzle 90.1

The price of a 1TB hard drive of a certain brand, when last checked, was ₹5,072 on online retail. To ensure that online retailers do not draw its customers away, an electronics store offers its 1TB hard drives at exactly the same price as they are available online

At the end of the month, the manager of the store first accounts for its sales with pen and paper, intending to enter the details later into an Excel sheet in the store’s main computer. Working at home, he is meticulous with his multiplications and additions (he uses a calculator, of course) and writes down the final numbers in the following format:

Sold ___ 1TB hard drives @ ₹5072 for a total of ₹______

When his wife calls him to dinner, he leaves the sheet of paper on the desk. In his absence, his infant child climbs up the desk, uncaps a bottle of ink, and spills it contents all over the desk.

When the manager returns to his desk, he finds to his dismay that part of what he had written down is now covered in ink. What he sees is the following:

Sold ___ 1TB hard drives @ ₹5072 for a total of ₹____264

The number of hard drives sold is totally covered in ink. The manager remembers that the total proceeds from 1TB hard drives were in six figures, but the first three of these digits have been inked out. He does not remember how many hard drives were sold, far less the six-digit amount received.

Every sale is, of course, individually accounted for in the computer records, but the store’s accounting system does not automatically total the number of items sold.

“Nothing to do,” the manager sighs, “but check the sales records again and count every 1TB hard drive sold, one by one.”

That will obviously take him a long time. Had he been strong at algebra, particularly with Diophantine equations, he could have set up a single equation in two variables and solved it in a few minutes.

Can you solve it for him?

Puzzle #90.2

I have a word of 10 letters.

1. One vowel appears twice.

2. The other 8 letters are 6 consonants and 2 vowels, none repeating.

3. The first five letters spell out a word that is an abbreviated form of the whole 10-letter word.

4. Letters 9–1–6–8–7–4–2 spell out a verb that means to request.

5. Letters 6–4­–5–1–7–3–2 spell out a verb that means to upgrade.

6. Letters 4–9–6–5–10–3–2 spell out a word that means a rejoinder

7. Letters 3–5–9–8–2–4 spell out a noun that means someone who works hard

8. Letters 10–9–1–6–8–2 spell out an adjective that means easy.

Easy, isn’t it?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 89.1

Hello Kabir,

The four aces were placed on top of the deck in the beginning. These cards become the top 4 cards of pile A initially. Then one card each from Pile B, Pile C and Pile D is placed on top of pile A in the first three rounds. In the fourth round, those three cards from the other piles are dealt down, and the original top 4 cards of Pile A (4 aces) are back again on top. Now, one ace each is distributed to the top of piles B, C and D. The fourth ace remains on top of pile A. This way, the top cards of all four piles end up being four aces.

— Sanjay S, Coimbatore

Sampath Kumar, also from Coimbatore, adds:

“I used to play this trick during my outings with friends. But I would add a lot of unnecessary steps, just to make it difficult for anyone to decipher it. For example, I would take 2 cards from B and place one each on top of C and D, then take 2 cards from C and place one each on top of B and D, and then 3 cards from A to B, C, D. No one had figured out the trick until now. Now that I have shared your puzzle with my friends and students, I have as good as shared this trick with the public. And that is for a very worthy puzzle.”

#Puzzle 89.2

Hi Kabir,

The probability of at least one card being red is 75% or 3/4. There are ways to solve this. The first is the classical mathematical solution: The probability of no red balls is 1/2 X 1/2 = 1/4, so the probability of at least one red ball is 1 –1/4 = 3/4. The second method uses common sense: If you take 2 balls, then you have the combinations BB, BR, RB, RR. As you can see, of the 4 possible outcomes, 3 outcomes have at least one red, so the probability = 3/4 or 75%.

— Akshay Bakhai, Mumbai

Solved both puzzles: Sanjay S (Coimbatore), Sampath Kumar V (Coimbatore), Akshay Bakhai (Mumbai), Kanwarjit Singh (Chief Commissioner of Income tax, retired), Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), Prof Anshul Kumar (Delhi), YK Munjal (Delhi), Raghunathan Ravindranathan

Solved #Puzzle 89.1: Sriram Prasad MI (Theni), Anil Khanna (Ghaziabad)

Solved #Puzzle 89.2: Ajay Ashok (Mumbai), Yadvendra Somra (Sonipat), Dr Vivek Jain (Sonipat)

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