How many digits?

How many digits can a sum or a difference have? Two 5-digit numbers can add to a 5-digit number (12,350 + 24,545 = 36,895), and one 5-digit number minus another can give a 5-digit number (48,952 − 24,547 = 24,405). But some answers are impossible. To find out, test the biggest and the smallest numbers.

Can two 4-digit numbers add up to a 6-digit number? The biggest 4-digit number is 9999, and 9999 + 9999 = 19,998: only 5 digits. So it never happens.

Can two 5-digit numbers add up to 18,500? The smallest 5-digit number is 10,000, and 10,000 + 10,000 = 20,000. Every such sum is at least 20,000, so 18,500 is impossible. In the same way, the biggest difference of two 5-digit numbers is 99,999 − 10,000 = 89,999, so 91,500 is impossible.

Can a 5-digit number minus a 2-digit number be a 3-digit number? The smallest answer is the smallest 5-digit number minus the biggest 2-digit number: 10,000 − 99 = 9901, still 4 digits. So never.

Some statements are only sometimes true. 5-digit + 5-digit gives a 5-digit number sometimes: 12,350 + 24,545 = 36,895 has 5 digits, but 60,000 + 50,000 = 1,10,000 has 6.