Answer: the number can be 9
Step-by-step explanation:
we want that
n + n - 1 + n-2 + .... + 2 + 1 = X
X must be a perfect square.
And the last number, 1, must be said by player 1, so n must be an odd number.
n + (n - 1) + (n -2) + ... + (n - n + 1) = n*n - (1 + 2 + 3 + .... + n)
and we know that:
(1 + 2 + 3 + 4 + ... + n) = n*(n + 1)/2
So we have:
x = n*n - n*(n + 1)/2 = n*n - n*n/2 - n/2 = n*n/2 - n/2 = n*(n + 1)/2
Now we want to find X such that is a perfect square and n must be an odd integer.
You can start giving different values for n until you reach a value of X that is a perfect square, for example, if you take n = 9, we have X = 36.
and 36 = 6*6
So if player 9, he will win always.
