Find a given number is reachable or not
There is number line ranging from -infinity to +infinity. Initially you are number 0. you can move in both directions(Moving left would decrease the number and moving right would increase the number). The condition is that in the ith step you have to iterate i numbers only, you can not stop in the middle. Now you have to tell that, whether a given number is reachable or not.
For example you can reach number 2 in following steps (0,1), (1,-1), (-1,2).
Also you need to provide the optimal solution to reach a given number if its reachable.
We can approach this problem by realizing that we can reach any integer n by making repeated jumps of size 1, alternating between moving to the left and moving to the right. Specifically, to reach n, we can make the following jumps:
- Move 1 unit to the right
- Move 2 units to the left
- Move 3 units to the right
- Move 4 units to the left
- Move 5 units to the right
- Move 6 units to the left
- and so on, continuing to alternate directions and increasing the size of the jump by 1 each time
This sequence of jumps will eventually reach n, since the sum of the jump sizes is 1 – 2 + 3 – 4 + 5 – 6 + … = 1 + (3 – 2) + (5 – 4) + (7 – 6) + … is an infinite series that diverges to infinity.
To find the optimal solution to reach a given number n, we can first check if it is reachable by computing the sum of the first k integers, where k is the smallest positive integer such that n is less than or equal to the sum. If n is not reachable, we can stop and return “not reachable”. Otherwise, we can find the sequence of jumps as follows:
- Start at 0
- Move 1 unit to the right
- Move 2 units to the left, unless we have already reached n
- Move 3 units to the right, unless we have already reached n
- Move 4 units to the left, unless we have already reached n
- Continue alternating between left and right, increasing the jump size by 1 each time, until we reach n.
This algorithm will terminate after at most 2n steps, since the sum of the jump sizes is at most 1 + 2 + … + n = n(n+1)/2. Therefore, it provides an optimal solution to reach a given number if it is reachable.
Your Answer
More puzzles to try-
What is the logic behind these ?
3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »Defective stack of coins puzzle
There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »Which clock works best?
Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »(Advanced) Cheryl’s Birthday Puzzle
Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers ...Read More »Five greedy pirates and gold coin distribution Puzzle
Five puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »Magical flowers!!
A devotee goes to three temples, temple1, temple2 and temple3 one after the other. In front of each temple, there ...Read More »Tuesday, Thursday what are other two days staring with T?
Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »How could only 3 apples left
Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »How Many Eggs ?
A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over ...Read More »HARD MATHS – How much faster is one train from other Puzzle
Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination ...Read More »Most Analytical GOOGLE INTERVIEW Question Revealed
Let it be simple and as direct as possible. Interviewer : Tell me how much time (in days) and money would ...Read More »Lateral thinking sequence Puzzle
Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »How did he know?
A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »Pizza Cost Math Brain Teaser
Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for $10. It turned out ...Read More »Which letter replaces the question mark
Which letter replaces the question markRead More »Which room is safest puzzle
A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »Richie’s Number System
Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »Srabon wanted to pass
The result of math class test came out. Fariha’s mark was an even number. Srabon got a prime!! Nabila got ...Read More »Become Normal!!
Robi is a very serious student. On the first day of this year his seriousness for study was 1 hour. ...Read More »Sakib Knows The Number!
Ragib: I got digits of a 2 digit number Sakib: Is it an odd? Ragib: Yes. Moreover, the sum of ...Read More »