To solve the puzzle, letβs define the blue number as \( b \) and the red number as \( r \). We have the following system of equations based on the clues provided:
1. \( 2b + r = 6 \) (from Clue 1)
2. \( 4b + 2r = 12 \) (from Clue 2)
We can simplify the second equation by dividing everything by 2:
\[
2b + r = 6
\]
Now, both equations are actually the same:
1. \( 2b + r = 6 \)
2. \( 2b + r = 6 \)
Since both equations represent the same line, they do not provide independent information. This means there are infinitely many solutions that satisfy this system because any value of \( b \) will produce a corresponding \( r \) that satisfies the equation.
### Conclusion
The answer is: **There are infinitely many solutions.**