The average of twelve numbers is $62$. The average of the last five numbers is $60$ and that of the first four numbers is $64$. The sixth number is $8$ less than the fifth number and $7$ less than the seventh number. What is the average of the sixth and seventh numbers?

Let's find the totals from the averages provided:
* Total sum of $12$ numbers = $12\times 62=744$
* Sum of the first $4$ numbers = $4\times 64=256$
* Sum of the last $5$ numbers = $5\times 60=300$

The remaining numbers are the $5^{\text{th}}$, $6^{\text{th}}$, and $7^{\text{th}}$ numbers.

$$\text{Sum of }(5^{\text{th}}+6^{\text{th}}+7^{\text{th}})=744-(256+300)=744-556=188$$

Let the $6^{\text{th}}$ number be $x$.

* The $6^{\text{th}}$ number is $8$ less than the $5^{\text{th}}$ number $⟹5^{\text{th}}\text{ number}=x+8$
* The $6^{\text{th}}$ number is $7$ less than the $7^{\text{th}}$ number $⟹7^{\text{th}}\text{ number}=x+7$

Setting up the algebraic equation:

$$(x+8)+x+(x+7)=188$$

$$3x+15=188⟹3x=173⟹x=\frac{173}{3}$$

We need the average of the $6^{\text{th}}$ and $7^{\text{th}}$ numbers:

$$\text{Average}=\frac{6^{\text{th}}+7^{\text{th}}}{2}=\frac{x+(x+7)}{2}=\frac{2x+7}{2}=x+3.5$$

$$\text{Average}=\frac{173}{3}+3.5=57.67+3.5=61.17$$