1. ## Positive Integers

12 positive integers are written in a row. The fourth number is 4, and the 12 number is 12. The sum of any three neighboring numbers is 333.Determine the twelve integers

2. Originally Posted by Dragon
12 positive integers are written in a row. The fourth number is 4, and the 12 number is 12. The sum of any three neighboring numbers is 333.Determine the twelve integers
You have the numbers: $\displaystyle A,B,C,4,D,E,F,G,H,I,J,12$

You know that: $\displaystyle 4+C+D=333$

Solve to get: $\displaystyle C=329-D$

You also know that: $\displaystyle 4+D+E=333$

Solve to get: $\displaystyle E=329-D$

Therefore: $\displaystyle C=E$

Using the same method you get that: $\displaystyle B=D$

So you know that: $\displaystyle C+4+D=333$

Substitute: $\displaystyle C+4+B=333$

Solve: $\displaystyle 4=333-B-C$

Note that: $\displaystyle A+B+C=333$

Solve: $\displaystyle A=333-B-C$

Therefore: $\displaystyle A=4$

Using the same methods above you'll find that:

$\displaystyle A=4=F=I$

$\displaystyle B=D=G=J$

$\displaystyle C=E=H=12$

Remember that: $\displaystyle D=333-4-C$

Substitute: $\displaystyle D=333-4-12$

Then: $\displaystyle D=317$

So the pattern is: $\displaystyle 4,317,12,4,317,12,4,317,12,4,317,12$