Playing with numbers: Mario came to class, wiped the blackboard and wrote...

​Mario came to class, wiped the blackboard and wrote on it the numbers 1 to 50, each exactly once. Kaja is going to play with these numbers. Exactly 49 times repeats the following procedure: chooses any two numbers on board, she erases both of them, and then on the board writes the value of difference. So if you erased the numbers x and y, back write the number | x - y |, i.e. the absolute value of the difference. During the game, it may happen that it will be written on the board several times the same number. For example, if Kaja right from the start erase numbers 6 and 9, writes on the blackboard No. 3. At that time, therefore, will be on board two threes. If then erased two threes, she wrote it instead on the board number 0.

A) Does Kaja know to play the game so that it in the end remained on the board number 14? If so, how? If not, why?

B) Find absolutely all possible values that can be the end of the game on the board.
Of course, substantiate your claim. (Can you really make each of these values and how?
And can you really not make any other already? Why?)

smartalec said:

​Mario came to class, wiped the blackboard and wrote on it the numbers 1 to 50, each exactly once. Kaja is going to play with these numbers. Exactly 49 times repeats the following procedure: chooses any two numbers on board, she erases both of them, and then on the board writes the value of difference. So if you erased the numbers x and y, back write the number | x - y |, i.e. the absolute value of the difference. During the game, it may happen that it will be written on the board several times the same number. For example, if Kaja right from the start erase numbers 6 and 9, writes on the blackboard No. 3. At that time, therefore, will be on board two threes. If then erased two threes, she wrote it instead on the board number 0.

A) Does Kaja know to play the game so that it in the end remained on the board number 14? If so, how? If not, why?

B) Find absolutely all possible values that can be the end of the game on the board.
Of course, substantiate your claim. (Can you really make each of these values and how?
And can you really not make any other already? Why?)

Click to expand...

Nice problem. Can you please tell us what you tried so some volunteer can guide you? Did you read the rules of the forum?

I know how it works but i m stuck with proving it

there are 25 odd numbees and 25 even and at the end always be last number odd but cant prove it

smartalec said:

I know how it works but i m stuck with proving it

Click to expand...

Have you tried anything to prove it. You claim you always get an odd number in the end so what seems to be happening? Show us what you have done so far?

For example when you make difference of two odd num. u get even when you make difference of two even num u get even.And when u make difference of odd and even u get odd. And now i tried something like this make 12 times this take 2 even and get 12 even numbers make 12 times odd and get next 12 even. And left is one odd and one even when i make difference i get odd. So i have 24 even numbers now one odd . Now i do 12 even numbers from 24 even and left is one odd. Next i do 6 even numbers from 12 even and left is one odd. Now i make 3 even from 6 even and left is one odd. Now u have 4 numbers 3 even 1 odd now u get one odd one even. And make last difference and only number what is written on blackboard is odd. U can make a lot of these selecting. And u always get odd at the end of game.