Question 9
On the blackboard a multiplication of 26 positive integers is written, that is, 26 integers separated by $\times$ signs. Lucía changes two of the $\times$ signs into $+$ signs and computes the result of the new expression. She repeats this procedure for every possible choice of two $\times$ signs in the initial expression. (The expression Lucía computes always has two $+$ signs and 23 $\times$ signs.) Of all the numbers Lucía obtains, exactly 115 are odd. Knowing that in the expression on the blackboard the odd numbers appear only in blocks of 2 and blocks of 3 (that is, there is no isolated odd number and there are never four consecutive odd numbers), compute how many of the 26 numbers on the blackboard can be odd.