Very cute question
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A frog can jump either right or left from his position.In his 1st jump he can go 1cm,2nd jump 2cm and in n-th jump it goes n cm. Prove that he can't back to his initial position in his 2019th jump.
Hmm..Hammer...Treat everything as nail
Re: Very cute question
I got a easy solution for this math . well , we have 1008 even jumps and 1009 odd jumps.
any kinds of sum of 1008 even numbers = random even number
any kinds of sum of 1009 odd numbers = random odd number
random even number + random odd number ≠ 0
so ,
he cant back to his initial position in his 2019th jump (proved )
any kinds of sum of 1008 even numbers = random even number
any kinds of sum of 1009 odd numbers = random odd number
random even number + random odd number ≠ 0
so ,
he cant back to his initial position in his 2019th jump (proved )
Last edited by Zafar on Sun Mar 14, 2021 7:14 pm, edited 1 time in total.
- Anindya Biswas
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Re: Very cute question
The problem with this argument is we actually have $1009$ even jumps and $1010$ odd jumps. $1+2+3+\dots+2019=\frac{2019\cdot2020}{2}$ which is even.Zafar wrote: ↑Sun Mar 14, 2021 6:46 pmI got a easy solution for this math . well , we have 1008 even jumps and 1009 odd jumps.
any kinds of sum of 1008 even numbers = random even number
any kind 1009 odd numbers = random odd numbers
random even number + random odd number ≠ 0
so ,
he cant back to his initial position in his 2019th jump (proved )
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
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Re: Very cute question
Note he can also jump right or left.Zafar wrote: ↑Sun Mar 14, 2021 6:46 pmI got a easy solution for this math . well , we have 1008 even jumps and 1009 odd jumps.
any kinds of sum of 1008 even numbers = random even number
any kind 1009 odd numbers = random odd numbers
random even number + random odd number ≠ 0
so ,
he cant back to his initial position in his 2019th jump (proved )
Hmm..Hammer...Treat everything as nail
Re: Very cute question
I dont know but when I read I saw 1009 even jumps . any kinds of sum means subtraction too . Well let me think now how to solve it now .Asif Hossain wrote: ↑Sun Mar 14, 2021 7:09 pmNote he can also jump right or left.Zafar wrote: ↑Sun Mar 14, 2021 6:46 pmI got a easy solution for this math . well , we have 1008 even jumps and 1009 odd jumps.
any kinds of sum of 1008 even numbers = random even number
any kind 1009 odd numbers = random odd numbers
random even number + random odd number ≠ 0
so ,
he cant back to his initial position in his 2019th jump (proved )
Re: Very cute question
well this sequence I found .
and the best approach of mine to solve it still now .
( its not a proof of course )
±(1-2-3+4+5-6-7+8.......2017-2018-2019+2020) =0
and here total numbers should be a product of 4.
but 2019 isnt a product of 4 .
then what ?
can you share your approaches ?
and the best approach of mine to solve it still now .
( its not a proof of course )
±(1-2-3+4+5-6-7+8.......2017-2018-2019+2020) =0
and here total numbers should be a product of 4.
but 2019 isnt a product of 4 .
then what ?
can you share your approaches ?
- Mehrab4226
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Re: Very cute question
The question doesn't seem correct.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
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-Henri Poincaré
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Re: Very cute question
Won't the frog change its direction after every jump so that it jumps on a rectangular or square path?
- Anindya Biswas
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Re: Very cute question
Not neccessarilyhriditapaul wrote: ↑Mon Mar 15, 2021 10:12 pmWon't the frog change its direction after every jump so that it jumps on a rectangular or square path?
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
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Re: Very cute question
i think something is wrong
here
here