a Numbers 1 through 50 are written on the board You can eith

a)

Numbers 1 through 50 are written on the board. You can either triple any number or replace any two numbers with their (positive) difference. This can be repeated as many times as you need. Can you end up with a single number which is equal to zero?

b)Numbers 1 through 50 are written on the board. You can add 1 to any two of them. This can be repeated as many times as you need. Can you make all the numbers equal?

Solution

b) no,

we prove it by contradiction ,

let us assume we can do it.

let us assume we add x times 1 to any two numbers .

and suppose y is number which is equal.

so the sum of all numbers = 50*y

which should also be equal to (1+ 2 +3 ..50) + x *2 = 2x +1275

so they must be equal.

2x + 1275 = 50 y

or -2 x + 50 y = 1275

2(25 y -x) = 1275

clearly this has no solution because LHS is even ,due to 2.

but RHS is odd.

hence our assumption is wrong.

so it can not be done.

Please ask a) part again