 Welcome friends, so this is another question of the day problem today I posted this question which says the numbers 1 through 20 are written in a row Now two players take turns putting plus signs and minus signs between the numbers and all such signs have been placed The resulting expression is evaluated that is addition and subtractions are performed The first player wins if the sum is even and the second wins if the sum is odd So the question is who will win and how now I have written these 20 numbers Not in a row, but you know in two rows But I have also shown some boxes where pluses or minus sign has to be placed now Let us say the first person, you know Puts this here Let us say plus sign between five and six and the second person puts minus sign like that and then again you know The first person puts minus here and the second person plus puts plus here and likewise he does it so we have to find out who will win and the player who wins if the sum is even and Second player wins wins if the sum is odd now if you see if you put any How many even numbers are there here? There are 10 even numbers. So basically there are 10 even numbers and and 10 odd numbers and odd Numbers, isn't it? now if you have 10 odd numbers then Let's take let's take case by case then even numbers are nothing but two four six eight ten likewise till 20 and Ten odd numbers are one three five seven likewise till 19, right? Now if you put any so basically what will happen? There will be either a plus sign or a minus sign In front of in all these digits, isn't it? Let us say for example when this guy was placing some sign between 15 and 16 So he has two choices either he will put plus 16 or he will put minus 16, right? so hence we can say that Along with all the numbers present here There will be either a plus sign or a minus sign, okay? So now let us take the case of even numbers now whether there is a plus sign ahead of two four six eight ten or a minus sign the result of all this sum together will always be even why because even plus even is even even plus even is equal to even and even minus even is also even Isn't it now come to the odd part of this so odd odd numbers are one three five seven now again There will be these numbers will be associated with either plus or minus sign now. We have ten odd numbers So basically ten odd numbers are there Then odd numbers are there that means I have five pairs of five pairs of odd numbers Is it it five pairs of odd numbers now if there are pairs of odd numbers? So whether whether two odd numbers are added or Subtracted the result is always even right so odd plus odd is equal to even and odd Minus odd is also even right and there are five such pairs So there will be five even numbers isn't it so we switch we saw in case one also there are the result is even so hence In case one there will be the result is even when this is nothing but all even numbers when all even numbers were considered the Some or the or the difference the total that is the calculation calculated value will be even and when all odd numbers were considered Then also there is a result of even right so even number is the result now even plus and again minus either even though adding the two even numbers and Adding that or subtracting the two even numbers will again get a Even number is it so hence what do we see? Irrespective of wherever the signs are put the result will always be even why again Some or differences of n even numbers will always result into a even number similarly some and differences of N odd numbers or five pairs of your number odd number. Sorry, we'll always result in a Even number so hence even plus minus even is even is it it? So food for thought for you will be what if we instead of taking first 20 20 numbers We would have taken let us say first 21 numbers that is after 20 we had 21 also then who would have won This is for you to do as an exercise. Thank you