## Task 4 - 060834

It is known that among 10 suitcases and 10 keys, each key fits exactly one suitcase, and for each suitcase, exactly one key fits. However, it is not known which key fits which suitcase.

Someone determines this by trial and error, where each trial consists of checking whether a specific key fits a specific suitcase or not. The order of the trials is chosen such that for each suitcase, once a trial has been conducted on it, exactly as many trials are performed until the correct key is found.

What is

a) the smallest

b) the largest

number of trials, after which it can happen that the correct key for each suitcase can be determined?
According to the task, at the suitcase on which the first test was conducted, exactly as many tests must be performed until the matching key is determined. This is the case at the earliest after one test and at the latest after 9 tests.

Then, a corresponding consideration must be carried out for the remaining 9 suitcases and keys, etc., up to 2 suitcases and keys. For these, all matching pairs can be determined with exactly one test. Therefore, the smallest number of tests of the desired type is $1+1+1+1+1+1+1+1+1=9$, and the largest number of tests of the desired type is $9+8+7+6+5+4+3+2+1=45$.