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Math Murder Mystery
by Total Gadha - Thursday, 1 February 2007, 09:26 PM
 

I found Ubhyankar Shastri immersed in his daily crossword when I reached his home in Srinivaspuri. The room was littered with books and coffee cups and I had to struggle to find a place to sit.

“Hi Devesh, How is our friend Raj keeping today?” he asked, putting aside the newspaper.

“Good heavens Ubhyankar! How do you know I met Raj. Were you spying on us?” Raj, or sub-inspector Rajnath Singh, was our mutual friend who often took Ubhyankar’s help to solve tricky cases. Since I was Ubhyankar’s confidant in many of his cases, Raj has got used to discussing criminal cases with me also.

“No no, nothing of the sort.” Ubhyankar started laughing, “You arrive every evening an hour earlier, therefore, I surmised you were stuck somewhere. When you walked in you had ‘Time’ magazine in your hand which you always buy from your favorite magazine shop in Nehru Place, and when you rubbed your feet on the doormat I saw you had wet mud on your shoes. I know it has rained in Kalkaji area near Nehru Place and today’s news stories in TV are full of murder committed in Kalkaji. As I know you couldn’t have gone to Kalkaji by yourself, someone must have taken you there. I’m positive it was our friend Raj.”

“That’s true. Raj knew you would be asleep till the evening (Ubhyankar had mastered the art of being completely nocturnal) therefore he took me along to Kalkaji to show the scene of the crime. The crime is pretty simple and I’m sure the TV stories would have given you the broad details. At 6:00 am, Mr. Nagpal Khanna, a collector of antique clocks, was discovered murdered in clocks room, his head having suffered injuries on account of being smashed against a showcase in which he kept his clocks. He was lying on the floor in a pool of blood and a rare agate and gold watch missing from his collection. Two other watches from the collection were lying on floor. Apparently, the thief had come in the midnight and had tried to steal the precious watch. Unfortunately, Mr. Khanna had woken up and had confronted the thief. A scuffle ensued and resulted in Mr. Khanna’s demise.”

“Did you look at the time in the watches lying on the floor?” Ubhyankar asked.

“Knowing your methods, I had a close look at those watches. They were both broken and had stopped working. The time they showed were 3:08 am and 2:24 am. You cannot determine the time of murder by looking at those watches. It is famous in Khanna household that every clock except the stolen one deviates by a fixed number of minutes every hour. These minutes may be same or different for different clocks. Every night before sleeping, Mr. Khanna used to wind each clock at 10:00 pm to show correct time. And only he knew by how many minutes each clock deviates every hour.”

“Any suspect?”

“Three in fact. All three of them are clock dealers and are of suspicious characters. Only these three suspects know the value of the clock and would be able to sell it to a secret buyer. All three had offered to buy the clock from Mr. Khanna and then threatened him when he refused. Chandraswami, the first suspect, says he was watching a movie till 1:15 am and reached home by 1:50 am. His family confirms the time he reached home. Saket Sharma, the second suspect, says he was in a party and got free by 2:10 am. He reached his home by 2:40 am. His family and the people in the party confirm the time. Rakesh Mahajan, the third suspect, was at Hotel Park Royale till 3:10 am. He reached his home at around 3:50 am. This has also been confirmed. Any of these three people could have committed the crime as they were near Mr. Khanna’s house. But we don’t know the time of crime and their alibi are perfect. That’s why Raj needs your help.”

“My dear Devesh, I can solve this case even without the help of caffeine you know. Tell Raj to question Rakesh Mahajan and search his house if necessary.”

“I am afraid I don’t understand Ubhyankar,” I said, at my wit’s end.

“It’s a simple case of mathematical reasoning Devesh. There are only three cases possible with the stopped clocks that you found on the floor- one was fast and the other was slow, both of them were fast, or both of them were slow. Now let us examine the possibilities:

One clock gains and the other loses

Let one of the clock gain x minutes every hour and the other clock lose y minutes every hours. Therefore the difference in minutes between them would be x + y every hour. Let a hours have passed after 10:00 pm when the crime was committed. The time difference between the two clocks when they stopped was 44 minutes.

Þ ( x + y) ´ a = 44

Some of the values in the solution set are shown below, assuming x + y is an integer:

image

Both clocks gain or both clocks lose

Let the two clocks lose/gain x and y minutes every hour, and let a hours had passed since 10:00 pm when the crime was committed. Then,

Þ (x - y) ´ a = 44 (both gain)

Þ (x - y) ´ a = 44 (both lose)

You can see that only the third equation has a solution which satisfies the conditions of gain/lose and which also give a time (3:30 am) when a suspect was free. Therefore, the time was committed at 3:30 am by Rakesh Mahajan.

I will leave the extensive mathematical explanation to the readers, but it turned out that Ubhyankar was right. Rakesh Mahajan admitted to his crime when Raj confronted him.

As for me, I still don’t understand what Ubhyankar said. Do you?

Re: Math Murder Mystery
by flyto gmat - Friday, 25 May 2007, 08:14 PM
 

Hi,

Iam a little confused over this,

is this the right way to approach the prob?

In my opinion, from the data about the 3 suspects, it can be noted , that the murder could have occured during

1:15  to 1:50 (that is between 3 hrs and 4hrs from 10:00 PM  )

2:10 to 2:40 (that is netween 4hrs and 5hrs from 10:00 PM )

3:10 to 3:50 (thats between 5hrs and 6 hrs from 10:00 PM )

That is, we can consider 1:00 , 2:00, 3:00 and 4:00 as the limits.

I guess, the most easy to start with option would be to consider that the murder occured after 4 hrs. That is 2:00.

a. both clocks show more than 2:00. So Both have gained.

Clock A gained 24mins and B gained 68mins in 4 hrs.

4X = 24, X=6. 4Y=68, Y=17.

4(Y-X) = 4(17-6) = 44. True.

b. Consider murder occured at 3:00. That is after 5hrs. Clock A lost 36 mins and Clock B gained 8 mins in 5 hrs.

5X = 36, X=7.2. 5Y=8, Y=1.6

5(Y+X)=5(1.6+7.2)=5(8.8)=44

 Please let me know

 

Re: Math Murder Mystery
by Ranvijay Singh - Thursday, 2 August 2007, 03:30 PM
  Hi TG,

Possible time-slots for the murder:

Person                 Time-Slot
Chandraswami       1:15 AM - 1:50 AM [10:00 PM + (39/12 hours - 46/12 hours)]
Saket Sharma       2:10 AM - 2:40 AM [10:00 PM + (50/12 hours - 56/12 hours)]
Rakesh Mahajan    3:10 AM - 3:50 AM [10:00 PM + (62/12 hours - 70/12 hours)]

Saket Shrama can't be the murderer because the open interval (50/12, 56/12) doesn't have any number, which when multiplied by an integer (x + y) or (x - y), gives 44. We have one such number for Chandraswami from his slot: 12 * 44/12 = 44, and 44/12 belongs to the open interval (39/12, 46/12). Rakesh Mahajan also has one such number from his slot: 8 * 66/12 = 44, and 66/12 clearly belongs to the open interval (62/12, 70/12).

Hence, there are two possible solutions to the question, which are as follows:

#1 - Chandraswami at 1:40 AM [a = 44/12 = 11/3, (x - y) = 12]
10:00 PM + 11/3 hours = 1:40 AM (not 2:40 AM as given in the case study above).

Now, this can be the time of the murder only when both watches gain. i.e., x - y = 12.


Total Gain for Watch #1 = 3:08 AM - 1:40 AM = 88 minutes
Total Gain for Watch #2 = 2:24 AM - 1:40 AM = 44 minutes

Evidently, x = 2y

So, the equation can be written as 2y - y = 12 => y = 12 and hence x = 24

[Verfication: 10:00 PM + 11/3 hours + 24 * 11/3 minutes = 3:08 AM and similarly we can verify the time on the second watch also]


#2 - Rakesh Mahajan at 3:30 AM [a = 66/12 = 11/2, (x - y) = 8]
10:00 PM + 11/2 hours = 3:30 AM

So, this is possible only when both the watches lose time.

Total loss for Watch #1 = 3:30 AM - 3:08 AM = 22 minutes
Total loss for Watch #2 = 3:30 AM - 2:24 AM = 66 minutes

Therefore, x = 3y

Now, 3y - y = 8 => y = 4, and hence x = 12

[Verification: 10:00 PM + 11/2 hours - 11/2 * 4 = 3:08 AM. Similarly, we can verify the time on the second watch as well]

I think both these solutions satisfy all the conditions of the case let. Please let me know how can Chandraswami get away with the murder? smile


Vijay
Re: Math Murder Mystery
by Total Gadha - Saturday, 4 August 2007, 07:00 AM
  Hi Ranvijay,

Good work. And yes, you are correct. smile Small mistake in time calculation from my side. Will correct it. smile

Total Gadha
Re: Math Murder Mystery
by vishal bawa - Wednesday, 10 June 2009, 01:19 PM
  great dimaag!
Re: Math Murder Mystery
by Ravi Vaghanani - Monday, 26 August 2013, 04:39 PM
  good explanation TG...