Question 7

We need to count the number of segments (sticks) used by each digit.

Step 1 — Count sticks for each digit Let us count the sticks for each digit. We will use the given diagram. Let $S_d$ be the sticks for digit $d$. Here is a list of stick counts.

| Digit | Sticks ($S_d$) | |---|---| | 0 | 6 | | 1 | 2 | | 2 | 5 | | 3 | 5 | | 4 | 4 | | 5 | 5 | | 6 | 6 | | 7 | 3 | | 8 | 7 | | 9 | 6 |

Step 2 — Make any number using 24 sticks We need to find a number. The total number of sticks must be 24. Let us try a number with four digits. The average sticks per digit is 6. Digits 0, 6, and 9 use 6 sticks each. Let us use four '6's. The number of sticks for 6666 is: $S_6 + S_6 + S_6 + S_6$ $= 6 + 6 + 6 + 6$ $= 24$

$\boxed{\text{Number: } \mathbf{6666}}$

Step 3 — Find the biggest number using 24 sticks To make the biggest number, we need many digits. Digits with fewer sticks make more digits. Digit '1' uses the fewest sticks. It uses 2 sticks. Let us find the maximum number of digits. Maximum digits = Total sticks $\div$ Sticks per '1'. $24 \div 2$ $= 12$ So, we can make a 12-digit number. All digits must be '1'. Any other digit uses more than 2 sticks. Using other digits reduces the total number of digits. A number with more digits is always larger. So, the biggest number has twelve '1's.

$\boxed{\text{Biggest number: } \mathbf{111111111111}}$

Step 4 — Find the smallest number using 24 sticks To make the smallest number, we need few digits. Digits with more sticks make fewer digits. Digit '8' uses the most sticks. It uses 7 sticks. Let us find the minimum number of digits. Minimum digits = Total sticks $\div$ Sticks per '8'. $24 \div 7$ $= 3 \text{ with remainder } 3$ This means we need at least 4 digits. So, the smallest number will have 4 digits. Let the 4-digit number be $d_1 d_2 d_3 d_4$. We want $d_1$ to be the smallest possible digit. The first digit cannot be zero. Let us try $d_1 = 1$. $S_1 = 2$ sticks. Remaining sticks for $d_2 d_3 d_4$ is $24 - 2 = 22$. Maximum sticks for 3 digits is $3 \times S_8 = 3 \times 7 = 21$. We need 22 sticks. We cannot make 3 digits with 22 sticks. So, $d_1$ cannot be '1'.

Let us try $d_1 = 2$. $S_2 = 5$ sticks. Remaining sticks for $d_2 d_3 d_4$ is $24 - 5 = 19$. We need to find $d_2 d_3 d_4$ using 19 sticks. We want $d_2$ to be the smallest possible digit. Let us try $d_2 = 0$. $S_0 = 6$ sticks. Remaining sticks for $d_3 d_4$ is $19 - 6 = 13$. We need to find $d_3 d_4$ using 13 sticks. We want $d_3$ to be the smallest possible digit. Let us check stick counts for digits. $S_0=6, S_1=2, S_2=5, S_3=5, S_4=4, S_5=5, S_6=6, S_7=3, S_8=7, S_9=6$. We need $S_{d_3} + S_{d_4} = 13$. To make $d_3 d_4$ smallest, we try small $d_3$. If $S_{d_3} = 6$ (digit '0'), then $S_{d_4} = 13 - 6 = 7$. Digit '8' uses 7 sticks. So, $d_4 = 8$. This gives $d_3 = 0$ and $d_4 = 8$. So, the number is 2008. Let us check the total sticks for 2008: $S_2 + S_0 + S_0 + S_8$ $= 5 + 6 + 6 + 7$ $= 24$ This number uses exactly 24 sticks. This is the smallest possible number.

$\boxed{\text{Smallest number: } \mathbf{2008}}$

Answer

(i) 6666 (ii) 111111111111 (iii) 2008