Analysis of the problem:

We are given a four-digit number. Let's represent the number as ABCD, where A, B, C, and D are the digits of the number.

From the problem statement, we have the following conditions:

1. The sum of the first two digits is equal to that of the last two digits:
A + B = C + D

2. The sum of the first and last digit is equal to the third digit:
A + D = C

3. The sum of the second and fourth digit is twice of the other digit:
B + D = 2 * C

Solution:

Let's solve the problem step by step.

Step 1: Simplifying the equations

From equation 2, we have A = C - D

Substituting this value in equation 1, we get:
C - D + B = C + D

Simplifying further, we get:
2D = B

So, B is twice the value of D.

Step 2: Possible values of B and D

Since B and D are digits, they can take values from 0 to 9.

From the equation 2D = B, we can consider the following possible pairs of B and D:

- B = 0, D = 0
- B = 2, D = 1
- B = 4, D = 2
- B = 6, D = 3
- B = 8, D = 4

Step 3: Finding the values of A and C

From equation 2, we have A = C - D

Substituting the values of B and D from the possible pairs, we get:

- For B = 0, D = 0: A = C - 0 = C
- For B = 2, D = 1: A = C - 1
- For B = 4, D = 2: A = C - 2
- For B = 6, D = 3: A = C - 3
- For B = 8, D = 4: A = C - 4

Step 4: Finding the four-digit numbers

Now, let's substitute the values of A, B, C, and D in the four-digit number ABCD.

For each possible pair of B and D, we can find the corresponding values of A and C.

For example, if we consider B = 0 and D = 0, then A = C, and the four-digit number becomes AAAA.

Similarly, we can find the four-digit numbers for other possible pairs of B and D.

Step 5: Final answer

The possible four-digit numbers that satisfy the given conditions are:

- 0000
- 1110
- 2220
- 3330
- 4440

These are the possible solutions to the problem.

Summary:

To summarize, we are given a four-digit number, and we need to find the numbers that satisfy the given conditions. By simplifying the equations and considering the possible values of the digits, we can find the solutions. The possible four-digit numbers that satisfy the conditions are 0000, 111