## Courtesy at stake

### The frequencies

The bar graph of Figure 7 shows the frequency with which each dice appears in different non-transitive cycles. A finer analysis of the graph shows that bars with the same height appear in pairs. If $$n_{1}\leq n_{2}\leq...\leq n_{5}\leq n_{6}$$ and $$m_{1}\leq m_{2}\leq...\leq m_{5}\leq m_{6}$$ are the number of spots in each dice of such a pair, they are related by the formula $$n_{i}+m_{i}=7$$ for all $$i=1,...,6.$$

Fig 7

click the image to see an interactive version