2 □ 1 □ 6 □ 6 = 48. What operations can replace the boxes?

Steps to Solve

1. Identify the required operations We need to find the appropriate operators to replace the blanks in the equation $2 ; \square ; 1 ; \square ; 6 ; \square ; 6 = 48$.

2. Test possibilities using multiplication and addition It's reasonable to initially consider multiplication since 48 is quite a large number. Start with $6 \times 6$.

   $$ 6 \times 6 = 36 $$

3. Add the product to the result of other operations Now, we can calculate using $2$ and $1$ to approach $48$:

We hypothesize $2 \times 1 + 36$:

$$ 2 \times 1 + 6 \times 6 = 2 + 36 = 38 $$

This does not equal $48$. Next, we test a different combination.

4. Test using multiplication and subtraction Now, let’s try $2 \times 1 \times 6 \times 6$ directly.

   $$ 2 \times 1 \times 6 \times 6 = 2 \times 36 = 72 $$

This exceeds $48$.

5. Find a valid combination Through trial, we consider $2 \times 6 \times 6 \div 1$:

   $$ 2 \times 6 \times 6 = 72 \div 1 = 72 $$

This still does not produce $48$.

6. Use division to refine the operations Testing just the intermediate recombination:

If we compute $2 \times (6 + 6)$;

$$ 2 \times (6 + 6) = 2 \times 12 = 24$$

This is shorter than required.

7. Final testing and strategy To find the combinations, hypothetical multiplication and verification through additional testing with varied placements:

Let’s test $2 \times 6 \times 4 = 48$. The use of arithmetic combinations can start with integers directly yielding needed targets.

Testing $2 \times 6 \div 1 \times 6$ confirmed that using $2 \times 6 + 6 \times 6 = 72$ not fitting, drawing checks if proper integrations render these values.