Question 5

We will analyze the segment counts for each digit and apply the given rules.

Step 1 — Counting sticks for 42,019

Let us count the segments for each digit. A digit '0' uses 6 segments. A digit '1' uses 2 segments. A digit '2' uses 5 segments. A digit '3' uses 5 segments. A digit '4' uses 4 segments. A digit '5' uses 5 segments. A digit '6' uses 6 segments. A digit '7' uses 3 segments. A digit '8' uses 7 segments. A digit '9' uses 6 segments.

For the number 42,019, we sum the segments. Segments for 4: 4 sticks. Segments for 2: 5 sticks. Segments for 0: 6 sticks. Segments for 1: 2 sticks. Segments for 9: 6 sticks. Total sticks needed: $4 + 5 + 6 + 2 + 6$ $= 23$ The number 42,019 requires exactly 23 sticks.

Step 2 — Making a bigger number with 25 sticks

We start with 42,019, which uses 23 sticks. We need to add two more sticks. So, the new number must use $23 + 2 = \textbf{25 sticks}$. The new number must also be bigger than 42,019. We can change one digit to add 2 sticks. Or we can change two digits, each adding 1 stick.

Let us find changes that add 2 sticks to a single digit:

• Change '4' (4 sticks) to '6' (6 sticks). This adds 2 sticks. The number becomes 62,019. This is bigger than 42,019.
• Change '4' (4 sticks) to '9' (6 sticks). This adds 2 sticks. The number becomes 92,019. This is bigger than 42,019.
• Change '2' (5 sticks) to '8' (7 sticks). This adds 2 sticks. The number becomes 48,019. This is bigger than 42,019.
• Change '1' (2 sticks) to '4' (4 sticks). This adds 2 sticks. The number becomes 42,049. This is bigger than 42,019.

Let us find changes that add 1 stick to two different digits:

• Digit '4' (4 sticks) can become '5' (5 sticks). This adds 1 stick.
• Digit '0' (6 sticks) can become '8' (7 sticks). This adds 1 stick.
• Digit '1' (2 sticks) can become '7' (3 sticks). This adds 1 stick.
• Digit '9' (6 sticks) can become '8' (7 sticks). This adds 1 stick.

Let us combine these changes to make a bigger number:

• Change '4' to '5' and '1' to '7'. The number becomes 52,079. This is bigger than 42,019.
• Change '0' to '8' and '1' to '7'. The number becomes 42,879. This is bigger than 42,019.
• Change '4' to '5' and '9' to '8'. The number becomes 52,018. This is bigger than 42,019.

We have found several numbers. Some other numbers bigger than 42,019 are:

$\boxed{\text{62,019, 92,019, 48,019, 42,049, 52,079, 42,879, 52,018}}$

Step 3 — Finding the biggest number by inserting '1'

The original number is 42,019. We want to insert an extra digit '1'. This will create a 6-digit number. To make the number largest, we place the '1' in a position that results in the highest value. We compare the numbers formed by inserting '1' at different places.

Let us list the possible numbers:

• Insert '1' before '4': 142,019
• Insert '1' between '4' and '2': 412,019
• Insert '1' between '2' and '0': 421,019
• Insert '1' between '0' and '1': 420,119
• Insert '1' between '1' and '9': 420,119
• Insert '1' after '9': 420,191

We compare these numbers from left to right. The largest number is 421,019. This is formed by inserting '1' between '2' and '0'.

$\boxed{\text{Insert '1' between '2' and '0' to get 421,019.}}$

Step 4 — Other numbers by placing '1'

We listed all the unique numbers formed in Step 3. These are the numbers Preetham can make.

$\boxed{\text{142,019, 412,019, 421,019, 420,119, 420,191}}$

Answer

(i) The number 42,019 requires exactly 23 sticks. (ii) Some other numbers bigger than 42,019 are 62,019, 92,019, 48,019, 42,049, 52,079, 42,879, 52,018. (iii) He should place the digit '1' between '2' and '0' to get the biggest number, which is 421,019. (iv) The other numbers he can make are 142,019, 412,019, 420,119, 420,191.