The Rest of the gedanken experiment, the beginning is in chapter \nameref{chap:discussion} section~\vref{discussion:subsec:gedanken_experiment}.
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They knew, that this could not be possible. Based on the assumption that the dice is fair, the statistical test should reveal only as much percentage of significant cases as the significant level was set. Here it should be around 50 (corresponds to $1,000\cdot0.05$).

They still were arguing with each other as the morning had already begun.  As they could not sleep anyway without solving this problem, they decided to call a friend who was mathematician. Latter suggested to try once the Kruskal-Wallis-test because it can be applied on a group of distributions. And behold, their friend was right: the new test revealed for an $\alpha=5\%$ around 50 significant cases and as $\alpha$ was set to 0.3\% they counted 4 significant cases, which is near by 3. This is exactly what they expected from a good test, see also chapter \nameref{chap:results} section \vref{results:sec:simulations}.
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