A diagram shows an input that goes through two rules to reach an output. The first rule is unknown (marked with ?). The second rule is divide by 3. Find the missing rule and the final output for an in

The diagram shows an input, a first rule (unknown, marked with "?"), a second rule ($\div 3$), and a final output. We need to find the missing rule in the first box.
We have the following input-output pairs:

• Input 24, Output 6
• Input 39, Output 11
• Input 54, Output 16
• Input 18, Output 4
• Input 45, Output 13
• Input 12, Output (missing)

The second rule is "$\div 3$". To find the value before this rule was applied (the intermediate value), we multiply the output by 3.

• For Input 24, Output 6: Intermediate Value $= 6 \times 3 = 18$.
• For Input 39, Output 11: Intermediate Value $= 11 \times 3 = 33$.
• For Input 54, Output 16: Intermediate Value $= 16 \times 3 = 48$.
• For Input 18, Output 4: Intermediate Value $= 4 \times 3 = 12$.
• For Input 45, Output 13: Intermediate Value $= 13 \times 3 = 39$.

Now we look at the relationship between the original Input and the Intermediate Value:

• Input 24 $\to$ Intermediate Value 18
• Input 39 $\to$ Intermediate Value 33
• Input 54 $\to$ Intermediate Value 48
• Input 18 $\to$ Intermediate Value 12
• Input 45 $\to$ Intermediate Value 39 In each case, the Intermediate Value is 6 less than the Input.

$$24 - 6 = 18$$ $$39 - 6 = 33$$ $$54 - 6 = 48$$ $$18 - 6 = 12$$ $$45 - 6 = 39$$

Therefore, the missing rule in the first box is "$-6$".