Brain Teasers
Tweeto's Puzzle !!!
The other day I was surprised by the sudden arrival of my friend Tweeto. He had arrived that morning after a long and arduous journey by train. Although he was quite tired out by the sheer length of the journey, he never wanted to miss a chance to catch me off guard with my math. As soon as he came in, he lay down a puzzle before me which quite took me by surprise.
He said his train journey had taken him through six stations: Maver, Willow, Wonderland, Doku, Lemp, culminating in the Rudolf Railway terminal (named after my grandfather Rudolf, the world-famous RomanKachi), from where he'd taken a cab to my house. He was one amongst the few people in the train as it had chugged into Maver.
These were his observations :-
1) At each of the stations Maver, Willow, Wonderland, Doku, Lemp and Rudolf, twice as many people got in as those that got down.
2) Further, the number of people getting down at each of these stations were all prime numbers - one each in between 0 and 10 at Maver, in between 10 and 20 at Willow, and so on up to in between 50 and 60 at Rudolf in the order of the stations mentioned above.
3) Also, the difference between the number of people getting in at any two consecutive stations was at least 20.
4) Finally, the total number of passengers getting down in all six stations together was an even number.
He wanted to know how many people (including him) were on the train before it had reached the Maver railway station if there were 227 people on it when it chugged into Rudolf (227 includes the people getting in and out at Rudolf).
It didn't take me long to disappoint him by finding the answer. Can you recreate history by solving my dear friend Tweeto's puzzle?
He said his train journey had taken him through six stations: Maver, Willow, Wonderland, Doku, Lemp, culminating in the Rudolf Railway terminal (named after my grandfather Rudolf, the world-famous RomanKachi), from where he'd taken a cab to my house. He was one amongst the few people in the train as it had chugged into Maver.
These were his observations :-
1) At each of the stations Maver, Willow, Wonderland, Doku, Lemp and Rudolf, twice as many people got in as those that got down.
2) Further, the number of people getting down at each of these stations were all prime numbers - one each in between 0 and 10 at Maver, in between 10 and 20 at Willow, and so on up to in between 50 and 60 at Rudolf in the order of the stations mentioned above.
3) Also, the difference between the number of people getting in at any two consecutive stations was at least 20.
4) Finally, the total number of passengers getting down in all six stations together was an even number.
He wanted to know how many people (including him) were on the train before it had reached the Maver railway station if there were 227 people on it when it chugged into Rudolf (227 includes the people getting in and out at Rudolf).
It didn't take me long to disappoint him by finding the answer. Can you recreate history by solving my dear friend Tweeto's puzzle?
Answer
The observations given are :-1) At each of the stations Maver, Willow, Wonderland, Doku, Lemp and Rudolf, twice as many people got in as those that got down.
2) Further, the number of people getting down at each of these stations were all prime numbers - one each in between 0 and 10 at Maver, in between 10 and 20 at Willow, and so on up to in between 50 and 60 at Rudolf in the order of the stations mentioned above.
3) Also, the difference between the number of people getting in at any two consecutive stations was at least 20.
4) Finally, the total number of passengers getting down in all six stations together was an even number.
From observations 1 and 3, since the difference between the number of people getting in at any two consecutive stations was at least 20, it means that the difference between the number of people getting down at any two consecutive stations was at least 10.
Let us start with the people getting down at Rudolf.
From observation 2, the number of people getting down at each of these stations were all prime numbers - one each in between 0 and 10 at Maver, in between 10 and 20 at Willow, in between 20 and 30 at Wonderland, in between 30 and 40 at Doku, in between 40 and 50 at Lemp, and in between 50 and 60 at Rudolf.
Hence, the number of people getting down at Rudolf should be either 53 or 59.
If it is 53, then the number of people getting down at Lemp should be either 43 or 41, and that at Doku has to be 31 (keeping in mind the condition that the difference between the number of people getting down at any two consecutive stations was at least 10). However, we cannot get a prime number for Wonderland in between 20 and 30 with a difference of 10 from Doku then.
This means that the number of people getting down at Doku must be 37 and not 31. Then, the figures for Lemp and Rudolf have to be 47 and 59 respectively. Then, we get the number of people getting down at Wonderland as 23. Now the figure for Willow can be 11 or 13. But if it is 11, we are left with no prime number for Maver (with a difference of 10). Hence, the number of people getting down at Willow must be 13.
Now the number of people getting down at Maver can be either 2 or 3. If this figure is 2, condition 4 would be violated, since the total number of passengers getting down in all six stations together would be an odd number. Hence, the figure for Maver should be 3.
Listing all these figures we have
(1st column is Station, 2nd is Getting down, 3rd is Getting in)
Maver 3 6
Willow 13 26
Wonderland 23 46
Doku 37 74
Lemp 47 94
Rudolf 59 118
Total 182 364
Given that the number of people when the train chugged into Rudolf is 227, the number of people including Tweeto in the train before it reached Maver =
(Number of people in the train at Rudolf) - (Net number of people who got in at the six stations) =
(Number of people in the train at Rudolf) - (Number of people who got in at the six stations - Number of people who got out at the six stations) =
227 - (364 - 182) =
227 - 182 =
45 people
Thus, I disappointed Tweeto by giving him the answer as 45 people.
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