Question 6

We will use the provided 7-segment display images. We will count the sticks for each digit. We will then find ways to move exactly four sticks. This means four sticks are removed and four sticks are added. The total number of sticks must stay the same.

Step 1 — Constructing 63,890

Let us first write down the number 63,890. We use the 7-segment display for each digit. The digit 6 uses 6 sticks. The digit 3 uses 5 sticks. The digit 8 uses 7 sticks. The digit 9 uses 6 sticks. The digit 0 uses 6 sticks. The total number of sticks for 63,890 is $6 + 5 + 7 + 6 + 6$.

$6 + 5 + 7 + 6 + 6$

$= 30$

$\boxed{\text{The number 63,890 uses 30 sticks.}}$

Step 2 — Understanding the "rearrange four sticks" rule

Let us list the number of sticks for each digit.

• Digit 0 uses 6 sticks.
• Digit 1 uses 2 sticks.
• Digit 2 uses 5 sticks.
• Digit 3 uses 5 sticks.
• Digit 4 uses 4 sticks.
• Digit 5 uses 5 sticks.
• Digit 6 uses 6 sticks.
• Digit 7 uses 3 sticks.
• Digit 8 uses 7 sticks.
• Digit 9 uses 6 sticks.

When we rearrange exactly four sticks, it means:

1. We remove exactly 4 sticks from some digits.
2. We add these same 4 sticks to other digits.
3. The total number of sticks in the new number must be the same.
4. So, the sum of sticks removed must equal the sum of sticks added. Both sums must be 4.
5. The new number must be greater than 63,890.

Step 3 — Finding other numbers bigger than 63,890

Let the original number be 63,890. It uses 30 sticks. We need to find changes that remove 4 sticks and add 4 sticks.

Example 1: Making 88,780 Let us change the first digit 6 to 8. This needs +1 stick ($S(8) - S(6) = 7 - 6 = 1$). Let us change the second digit 3 to 8. This needs +2 sticks ($S(8) - S(3) = 7 - 5 = 2$). Let us change the third digit 8 to 7. This removes -4 sticks ($S(7) - S(8) = 3 - 7 = -4$). Let us change the fourth digit 9 to 8. This needs +1 stick ($S(8) - S(9) = 7 - 6 = 1$). The last digit 0 stays as 0. This is 0 stick change. Total sticks added: $1 + 2 + 1 = 4$. Total sticks removed: $4$. The total change in sticks is $1 + 2 - 4 + 1 + 0 = 0$. The new number is 88,780. This is greater than 63,890.

Example 2: Making 98,788 Let us change the first digit 6 to 9. This is 0 stick change ($S(9) - S(6) = 6 - 6 = 0$). Let us change the second digit 3 to 8. This needs +2 sticks ($S(8) - S(3) = 7 - 5 = 2$). Let us change the third digit 8 to 7. This removes -4 sticks ($S(7) - S(8) = 3 - 7 = -4$). Let us change the fourth digit 9 to 8. This needs +1 stick ($S(8) - S(9) = 7 - 6 = 1$). Let us change the fifth digit 0 to 8. This needs +1 stick ($S(8) - S(0) = 7 - 6 = 1$). Total sticks added: $2 + 1 + 1 = 4$. Total sticks removed: $4$. The total change in sticks is $0 + 2 - 4 + 1 + 1 = 0$. The new number is 98,788. This is greater than 63,890.

Example 3: Making 68,788 Let us keep the first digit 6 as 6. This is 0 stick change. Let us change the second digit 3 to 8. This needs +2 sticks ($S(8) - S(3) = 7 - 5 = 2$). Let us change the third digit 8 to 7. This removes -4 sticks ($S(7) - S(8) = 3 - 7 = -4$). Let us change the fourth digit 9 to 8. This needs +1 stick ($S(8) - S(9) = 7 - 6 = 1$). Let us change the fifth digit 0 to 8. This needs +1 stick ($S(8) - S(0) = 7 - 6 = 1$). Total sticks added: $2 + 1 + 1 = 4$. Total sticks removed: $4$. The total change in sticks is $0 + 2 - 4 + 1 + 1 = 0$. The new number is 68,788. This is greater than 63,890.

Example 4: Making 88,248 Let us change the first digit 6 to 8. This needs +1 stick ($S(8) - S(6) = 7 - 6 = 1$). Let us change the second digit 3 to 8. This needs +2 sticks ($S(8) - S(3) = 7 - 5 = 2$). Let us change the third digit 8 to 2. This removes -2 sticks ($S(2) - S(8) = 5 - 7 = -2$). Let us change the fourth digit 9 to 4. This removes -2 sticks ($S(4) - S(9) = 4 - 6 = -2$). Let us change the fifth digit 0 to 8. This needs +1 stick ($S(8) - S(0) = 7 - 6 = 1$). Total sticks added: $1 + 2 + 1 = 4$. Total sticks removed: $2 + 2 = 4$. The total change in sticks is $1 + 2 - 2 - 2 + 1 = 0$. The new number is 88,248. This is greater than 63,890.

Answer

(i) The number 63,890 is formed by displaying the digits 6, 3, 8, 9, and 0 using 7-segment displays. The digit 6 uses 6 sticks, 3 uses 5 sticks, 8 uses 7 sticks, 9 uses 6 sticks, and 0 uses 6 sticks. (ii) Other numbers bigger than 63,890 that can be made by rearranging exactly four sticks are: 88,780 98,788 68,788 88,248