Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Question

Homi · Accepted Answer

Understand the Question

The original number has 20 digits in total.
Striking out $10$ digits leaves a 10-digit number ( $20 - 10 = 10$ ).
To make the resulting number as large as possible, we maximize the digits from left to right (starting with the highest place value) using a greedy approach, while always ensuring enough digits remain to complete a 10-digit number.

Step 1 · Find the First Digit

To leave at least $9$ digits for the remaining places, the first digit must be chosen from the first $20 - 9 = 11$ digits:

$12345123451$

The largest digit in this section is 5 (at the $5^{\text{th}}$ position).

Strike out the $4$ digits before it: $1, 2, 3, 4$ .

Step 2 · Find the Second Digit

We need $8$ more digits after the second digit. From the remaining $15$ digits, choose the second digit from the first $15 - 8 = 7$ digits:

$1234512$

The largest digit is 5 (at the $5^{\text{th}}$ position of this remaining part).

Strike out the $4$ digits before it: $1, 2, 3, 4$ .

Total digits struck out so far: $4 + 4 = 8$

Step 3 · Find the Third Digit

We need $7$ more digits after the third digit. From the remaining $10$ digits, choose the third digit from the first $10 - 7 = 3$ digits:

$123$

The largest digit is 3 (at the $3^{\text{rd}}$ position of this part).

Strike out the $2$ digits before it: $1, 2$ .

Total digits struck out so far: $8 + 2 = 10$

Step 4 · Form the Final Number

Since exactly $10$ digits have been struck out, no more digits can be removed. Keep all remaining $7$ digits:

$4512345$

Combining the chosen digits ( $5, 5, 3$ ) with the remaining digits ( $4, 5, 1, 2, 3, 4, 5$ ):

$5534512345$

Answer

$5534512345$

Common Mistakes

Exceeding the Strikeout Limit: Striking out digits to reach the next $5$ without realizing it would exceed the allowed limit of $10$ deletions, leaving fewer than $10$ total digits.
Not Working Left-to-Right: Trying to keep larger digits near the end rather than prioritizing the most significant place values on the far left.

Question 16