Large Numbers Around Us | A

Question 6

Make or write the number 63,890.
Starting with 63,890, rearrange exactly four sticks and make a bigger number. One example is 88,078. What other numbers bigger than 63,890 can you make in this way?

Solution

Understand the Question

Each digit is formed using matchsticks in a standard 7-segment digital display format.
The original number 63,890 uses a total of $30$ sticks.
Rearranging exactly four sticks means removing $4$ sticks from some digits and placing those same $4$ sticks onto other digits so that the total number of sticks remains $30$ .
The resulting number must be strictly greater than 63,890.

(1) Make or write the number 63,890.

Step 1 · Construct 63,890 and Count Sticks

Diagram 1

Sticks used for each digit:

Digit 6: $6$ sticks
Digit 3: $5$ sticks
Digit 8: $7$ sticks
Digit 9: $6$ sticks
Digit 0: $6$ sticks

Total number of sticks used: $6 + 5 + 7 + 6 + 6 = 30$

Answer

(1) The number 63,890 is formed using $30$ sticks.

(2) Starting with 63,890, rearrange exactly four sticks and make a bigger number. One example is 88,078. What other numbers bigger than 63,890 can you make in this way?

Step 1 · Stick Count per Digit and Rearrangement Rule

Number of sticks required for digits $0$ to $9$ :

0: $6$ sticks
1: $2$ sticks
2: $5$ sticks
3: $5$ sticks
4: $4$ sticks
5: $5$ sticks
6: $6$ sticks
7: $3$ sticks
8: $7$ sticks
9: $6$ sticks

Rule for rearranging 4 sticks:

Remove exactly $4$ sticks from one or more digits.
Add these $4$ sticks to other digits.
The resulting number must be greater than $63{,}890$ .

Step 2 · Form Other Numbers Bigger than 63,890

Let $S(d)$ denote the number of sticks in digit $d$ .

Making 88,780:

First digit: $6 \to 8 \implies S(8) - S(6) = 7 - 6 = +1$
Second digit: $3 \to 8 \implies S(8) - S(3) = 7 - 5 = +2$
Third digit: $8 \to 7 \implies S(7) - S(8) = 3 - 7 = -4$
Fourth digit: $9 \to 8 \implies S(8) - S(9) = 7 - 6 = +1$
Fifth digit: $0 \to 0 \implies 0$

Sticks added $= 1 + 2 + 1 = 4$ , Sticks removed $= 4$ . $\text{Net change} = 1 + 2 - 4 + 1 + 0 = 0$ $88{,}780 > 63{,}890$

Making 98,788:

First digit: $6 \to 9 \implies S(9) - S(6) = 6 - 6 = 0$
Second digit: $3 \to 8 \implies S(8) - S(3) = 7 - 5 = +2$
Third digit: $8 \to 7 \implies S(7) - S(8) = 3 - 7 = -4$
Fourth digit: $9 \to 8 \implies S(8) - S(9) = 7 - 6 = +1$
Fifth digit: $0 \to 8 \implies S(8) - S(0) = 7 - 6 = +1$

Sticks added $= 2 + 1 + 1 = 4$ , Sticks removed $= 4$ . $\text{Net change} = 0 + 2 - 4 + 1 + 1 = 0$ $98{,}788 > 63{,}890$

Making 68,788:

First digit: $6 \to 6 \implies 0$
Second digit: $3 \to 8 \implies S(8) - S(3) = 7 - 5 = +2$
Third digit: $8 \to 7 \implies S(7) - S(8) = 3 - 7 = -4$
Fourth digit: $9 \to 8 \implies S(8) - S(9) = 7 - 6 = +1$
Fifth digit: $0 \to 8 \implies S(8) - S(0) = 7 - 6 = +1$

Sticks added $= 2 + 1 + 1 = 4$ , Sticks removed $= 4$ . $\text{Net change} = 0 + 2 - 4 + 1 + 1 = 0$ $68{,}788 > 63{,}890$

Making 88,248:

First digit: $6 \to 8 \implies S(8) - S(6) = 7 - 6 = +1$
Second digit: $3 \to 8 \implies S(8) - S(3) = 7 - 5 = +2$
Third digit: $8 \to 2 \implies S(2) - S(8) = 5 - 7 = -2$
Fourth digit: $9 \to 4 \implies S(4) - S(9) = 4 - 6 = -2$
Fifth digit: $0 \to 8 \implies S(8) - S(0) = 7 - 6 = +1$

Sticks added $= 1 + 2 + 1 = 4$ , Sticks removed $= 2 + 2 = 4$ . $\text{Net change} = 1 + 2 - 2 - 2 + 1 = 0$ $88{,}248 > 63{,}890$

Answer

(2) Other numbers bigger than $63{,}890$ are 88,780, 98,788, 68,788, and 88,248.

Common Mistakes

Unequal Stick Movements: Moving fewer or more than exactly $4$ sticks. The total count of sticks removed must equal $4$ , and the total count of sticks added must equal $4$ .
Forming a Smaller Number: Ensuring the new number is greater than $63{,}890$ ; modifying leading digits to smaller values will result in a smaller number instead.
Incorrect Digit Stick Count: Miscounting the number of segments for digits in 7-segment displays (e.g., mistaking $7$ as using $2$ sticks instead of $3$ , or $3$ as using $6$ instead of $5$ ).

Question 6

More questions in A