<p>Your question has several aspects and it is unclear what statistical topic/problem you are aiming at with the question. You should narrow it down, or at least the problem statement could be more concise/isolated. (if it is about how to incorporate those many aspects into the problem then this can be done, but currently it is unclear what the basic problem is)</p>
<h3>1</h3>
<p>Your question has a big problem here</p>
<blockquote>
<p>as many cards as possible by opening as few boxes as possible</p>
</blockquote>
<p>it is already not clear who wins even when the timers working correctly. (I guess that you meant the person with the most cards wins, and the 'few boxes as possible' is the strategy to do it, but you should clarify that)</p>
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<h3>2</h3>
<p>The problem is not complete (as with many practical/story statistics problems), but it is too abstract (no clear context) that allows people to fill in the gaps.</p>
<blockquote>
<p>...the timers for rooms B and C had not been set up accurately ... we already need to decide the winner on the basis of the present results.</p>
</blockquote>
<p>There is no single solution to this problem and there will be different approaches possible. In the current state it is much too broad and it is unclear what the culprit is of your long question (what is the key problem/issue?). How should answers decide on that? It would help if you would give more guidelines what is desired and especially it would be nice if the question could be made to focus on a smaller problem or when the task/objective is mvd more clear.</p>
<hr />
<p>To me your question reads as</p>
<p><em>alice and bob are doing a beauty contest, they have to dress up within 1 hour, alice has three pretty flowers in her hair and bob has two nice cufflinks, but bob finished within 50 minutes due to an error with the clock. Who wins?</em></p>
<p>It is very vague to decide who wins and the underlying statistical problem, if there is any, is not exposed in such complex question.</p>
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<p>Possibly this question (given to you as an exercise?)  is about <a href="https://en.wikipedia.org/wiki/Pareto_efficiency" rel="nofollow noreferrer">Pareto efficiency</a>, or something related. Then it is meant to make you think and realize that: if there is no single person with both the least boxes and most cards, then there is no clear solution possible when you do not make a decision how to differentiate. (and the problem with the timers is adding more spice into the mix and allows you to ponder about different objectives, fairness/stability/mean utility/etc., that can be optimized).</p>
<p>But that efficiency and objectives is speculation about your question. It would greatly help if you explain the background of your question, where it comes from and why you try to answer it, in order for people here to guess the meaning of the questions and such that they do not need to speculate.</p>
