MathDB

To open the magic chest, one needs to say a magic code of length

n

consisting of digits

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Each time Griphook tells the chest a code it thinks up, the chest's talkative guardian responds by saying the number of digits in that code that match the magic code. (For example, if the magic code is

0423

and Griphook says

3442,

the chest's talkative guard will say

1

). Prove that there exists a number

k

such that for any natural number

n \geq k,

Griphook can find the magic code by checking at most

4n-2023

times, regardless of what the magic code of the box is.

Problem Statement