Calvin was asked to evaluate $37 + 31 \times a$ for some number $a$. Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated $37 \times 31 + a$ instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations. For what value of $a$ could this have happened? 

[i]Ray Li.[/i]
1. Let's denote the correct expression Calvin was supposed to evaluate as \( 37 + 31 \times a \).
2. Calvin mistakenly evaluated \( 37 \times 31 + a \) instead. 
3. According to the problem, Calvin still arrived at the correct answer. Therefore, we set up the equation:
   \[
   37 + 31 \times a = 37 \times 31 + a
   \]
4. First, calculate \( 37 \times 31 \):
   \[
   37 \times 31 = 37 \times (30 + 1) = 37 \times 30 + 37 \times 1 = 1110 + 37 = 1147
   \]
5. Substitute \( 1147 \) into the equation:
   \[
   37 + 31a = 1147 + a
   \]
6. To isolate \( a \), subtract \( a \) from both sides:
   \[
   37 + 31a - a = 1147
   \]
   Simplify the left-hand side:
   \[
   37 + 30a = 1147
   \]
7. Subtract 37 from both sides to isolate the term with \( a \):
   \[
   30a = 1147 - 37
   \]
   Simplify the right-hand side:
   \[
   30a = 1110
   \]
8. Divide both sides by 30 to solve for \( a \):
   \[
   a = \frac{1110}{30} = 37
   \]

The final answer is \( \boxed{37} \)