Given information:
- The carpenter allows a discount of 14% to his customer.
- The carpenter still gains 20% from the sale.

To find:
- The marked price of the center table.

Step-by-step solution:

Step 1: Calculate the selling price after the discount.
Let's assume the marked price of the center table is 'x' dollars.

The discount allowed by the carpenter is 14%, so the customer pays 100% - 14% = 86% of the marked price.

Selling price = 86% of the marked price
= (86/100) * x
= 0.86x dollars

Step 2: Calculate the cost price of the center table.
The carpenter still gains 20% from the sale, which means the selling price is 120% of the cost price.

Selling price = 120% of the cost price
= (120/100) * cost price
= 1.2 * cost price

Since we know the selling price is 0.86x dollars, we can equate it to the cost price:

1.2 * cost price = 0.86x dollars

Step 3: Find the marked price.
To find the marked price, we need to solve the equation from step 2 for x.

1.2 * cost price = 0.86x dollars

Dividing both sides of the equation by 0.86:

(1.2 * cost price) / 0.86 = x

Therefore, the marked price of the center table is (1.2 * cost price) / 0.86 dollars.

Step 4: Substitute the given values to find the marked price.
Given that the carpenter gains 3910 dollars from the sale, we can substitute this value for the cost price in the equation from step 3:

Marked price = (1.2 * 3910) / 0.86
= 4692.48 dollars

Therefore, the marked price of the center table is $4692.48.

Conclusion:
The marked price of the center table is $4692.48. The carpenter allows a 14% discount to the customer and still gains 20% from the sale, resulting in a selling price of $3910.