If a team's ratio of wins to losses increases from  to  by winning six games, they must have 12 wins initially and 3 losses. 

The rule is as follows: add 2 to the previous number then multiply times 2.

The answer is (16 + 2) * 2 = 36.

In order to find the missing number (X), you must first identify the pattern in the set of numbers. Upon a glance, you can determine that the pattern starts with 4 and then the next number is 2n+1, with n being the preceding number in the set.

For example, let's start with 4. Multiplying 2*4+1 gives us 9, the next number in the set. 

Therefore, the missing number in the set is .

The lower right number of each box is the sum of the other three numbers, which are found as follows:

The upper left numbers are simply the natural numbers in order;

The upper right numbers are the powers of 2, beginning with ;

The lower left numbers are the terms of the Fibonacci sequence, which is formed by setting the first two terms equal to 1 and each subsequent term equal to its predecessors.

Therefore, the ninth box in the sequence includes the number 9 as its upper left entry, and  as its upper right entry. The lower left entry is the ninth Fibonacci number, which can be found as follows:

The lower right entry is the sum of the other entries:

There will be 15 gaps of  long but 16 tick marks because there will be a tick mark on each end of the ruler.

Let x equal the smallest of the four numbers. Therefore:

Therefore the four odd numbers are 7, 9, 11, and 13. Since all are prime except 9, three of the numbers are prime.

We know that the remainder, , must be less than  by the definition of remainder. Therefore our only choices are , , or . We can test each of these cases. 

If   would be divisible by , which we said is not true. 

If : Try . Then . When we divide  by , we have a remainder of . This works! 

If : Try . Then . When we divide this by , we have a remainder of . Thus, our remainders are not equal. 

Thus, . 

Let's look at a small example, the numbers between 1 and 3 including all endpoints. We have 1, 2, and 3. If we subtract our numbers, we get 2. We need to add one more to get all the desired numbers. We can follow the same process with our larger numbers. 

Take,

 .

Then, adding 1,  

.

Therefore, our answer is 

To count the number of pages in between these two page numbers, we want to subtract  and then add . We have to add  because if we don't, we end up not counting either the first or last page of the selection. 

Therefore, our answer is 

An important consideration whenever you're asked for the number of integers between two other numbers is to determine whether the two "end points" are included. Here the word "inclusive" tells us that we need to count both -8 and 17 in our calculation.

A helpful tip for these questions: if you're including both end points, you can calculate by using the range (greatest value minus smallest value) and adding 1 to account for the inclusive set.  (If you need to prove that to yourself, just use "how many integers between 1 and 3 inclusive" - clearly the set is 1, 2, 3 for three integers, but 3-1 = 2, so you can see you need to add one more to get the answer).

For this question, that leaves us with 17 - (8) = 25 as the range, and if you add 1 to that for "inclusive" you get the correct answer, 26.

Another way to look at this one is to chop it up: between -8 and 17 there are 8 negative integers (-1 through -8), 17 positive integers (1 through 17), and 0 as one more integer.  8 + 17 + 1 = 26.