Countdown Numbers

The answer is $\boxed{1977}$ . To see that this is achievable, consider the following sequence (with target number 999):

• $10$ [New best]
• $10 + 10 = 20$ [New best]
• $20 * 100 = 2000$
• $2000 + 2 = 2002$
• $2002 - 25 = 1977$ [New best]

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First, it should be rather intuitive that it is always optimal to choose 999 as the target number. Here's some explanation if that's not clear to you: note that for any number larger than 1000, the target number will be farther away if it is less than 999, and thus it will be harder to achieve that number. This means that if we seek to achieve a number larger than 1977, it will always be best to use 999 as the target number.

Now, suppose we seek to have the closest number display read 1978. Then to make sure that this number is possible, the largest number we can have on the display before it that is smaller than it has to be $999-(1978-999)-1=19$ , since anything larger would be as close or closer to the target number 999. Let the last of these "small" numbers that appears on the display be $x$ . Then the number that appears after $x$ is at most $x*100$ or $x+100$ ; since we know $x \le 19$ , the number on the display after $x$ must be at most $19*100=1900 < 1978$ , so $1978$ will never be on the "closest number" display, as desired.