Antal numbers 100 matchboxes from 1 to 100 and puts a random number of matches in each. Bea randomly selects 15 boxes, and Antal counts the matches in them (without Bea seeing this), and tells her whether the total number of matches in the 15 boxes is even or odd. Bea can repeat this questioning step as many times as she wants. Can Bea figure out whether there is an even or odd number of matches in the box numbered 1, and if so, what is the minimum number of steps required for this?
Bea needs 3 questions to surely determine whether there is an even or odd number of matches in the 1st box. For example, the following three questions are sufficient:

I. Boxes 1-15, II. Boxes 1-8 and 16-22, III. Boxes 1 and 9-22.

By adding the parities of the answers according to the known rules (even + even = odd + odd = even, etc.), we get the parity of the number of matches in the 1st box, since in the sum of the three answers, the content of the 1st box appears 3 times, while the content of the 2nd to 22nd boxes appears twice, so this sum differs from the number of matches in the 1st box by an even number.

We also need to show that fewer than three questions are not sufficient. If the 1st box does not appear in any of the questions, then by adding one match to it, the answers do not change, but the parity of the number of matches in the 1st box does, so it cannot be determined whether there is an even or odd number of matches in it.

It is easy to see that one question, or two identical questions, or two questions with no common elements are not sufficient, since in these cases, by adding one match to the 1st box and one to a box that appears with it in a question, the answers do not change, but the parity of the number of matches in the 1st box does.

If none of the above cases apply, let

A: be the set of boxes that appear only in the first question,

B: be the set of boxes that appear only in the second question, and

C: be the set of boxes that appear in both questions.

According to our assumption, none of A, B, and C can be an empty set, and at least one of them must contain the 1st box. If we now add one match to the 1st box and one to a box in each of the sets that do not contain the 1st box, the answers to the two questions do not change, but the parity of the number of matches in the 1st box does.

By comparing our results, we conclude that Bea can determine the parity of the number of matches in the 1st box, and she needs 3 questions to do so.

Based on the work of András Gefferth (Bp., Fazekas M. Gyak. Gimn., II. o. t.)