Question 7

Step 1 · Construct a 4-Digit Number

Using four digits that each require $6$ sticks (such as $6$):

Therefore, a valid number is $6666$.

Step 1 · Maximize Number of Digits

To form the largest number, maximize the count of digits by using the digit with the minimum stick requirement ($1$, which requires $2$ sticks):

$\text{Number of digits} = \dfrac{24}{2} = 12$

Placing twelve $1$s gives the largest possible number:

$111111111111$

Step 1 · Determine the Minimum Number of Digits

To make the smallest number, minimize the total number of digits. Since digit $8$ uses the most sticks ($7$ sticks):

$24 \div 7 = 3 \text{ with remainder } 3$

Since $3$ digits can use at most $3 \times 7 = 21$ sticks, a minimum of $4$ digits ($d_1 d_2 d_3 d_4$) is required.

Step 2 · Determine the Digits to Minimize the Value

To minimize $d_1 d_2 d_3 d_4$, make the leading digits as small as possible:

1. First digit ($d_1$):

   • Cannot be $0$.
   • If $d_1 = 1$ ($2$ sticks), remaining sticks $= 24 - 2 = 22$. But $3$ digits can hold at most $3 \times 7 = 21$ sticks, which is impossible.
   • If $d_1 = 2$ ($5$ sticks), remaining sticks $= 24 - 5 = 19$.

2. Second digit ($d_2$):

   • Try smallest possible digit $d_2 = 0$ ($6$ sticks).
   • Remaining sticks for $d_3 d_4 = 19 - 6 = 13$.

3. Remaining digits ($d_3, d_4$):

   • To minimize $d_3$, choose $d_3 = 0$ ($6$ sticks).
   • Remaining sticks for $d_4 = 13 - 6 = 7$, which corresponds to digit $8$.

Checking the total sticks for $2008$: