1} For pattern 1, we figured out that 2x3+3=9 or 3 squared. The equation is A x B + B. B squared is the answer to the equation. To confirm we tried 5x6+6=36 or 6 squared as equation 1. We also tried the other 2 equations and concluded that all the equations are B squared as stated above.
2} For pattern 2, we tried the numbers 1,2,3. We followed the instructions so we ended with 4-1x3=1. We also tried 9,10,11 and ended with the number 1,again. All the answers for equations equal 1.
3} For pattern 3, we figured out that A x A - B x B = N. The number in between A and B is multiplied by 4 to make N. For example: 3 x 3 - 1 x 1 = 8. So the number between 3 and 1 is 2 so 2 x 4 = 8. Another example is: 10 x 10 - 8 x 8 = 36. If we apply the same solution you will see that it will work.
4} We used a substitute for N, such as 2,3,4 and 5. We applied the substitutes to the equations and all the solutions always equal to 2. For example: 3(3+1)-(3-1)(3+2)=2. 3 is the substitute for N.
5} The pattern is what Charlie quoted,"If you multiply two numbers that differ by 2, and then add one, the answer is always the square of the number between them!" 3 x 5 + 1= 16 or 4 squared. 5 x 7 + 1 is 36 or 6 squared.
6} For pattern 6, the result equals the middle number squared, then times by 2. For example: 1,2,3. 1 x 2=2, 2 x 3=6 therefore 6 times 2 equals 8. Another example is 50,51 and 52. 50 x 51= 2550, 51 x 52= 2652. Therefore 2550 + 2652=5202 .51 squared then times 2 equals 5202.