$ T1: Bell number $
Questions directly transfer equation gives a similar $ Fibonacci \ sequence $, apparently can think of matrix quickly optimize power
However, we can only find answers to the modulo small primes answers
Modulus is the product of exactly $ 5 $ small primes
Finally, out of $ crt $ enough
$ T2: crossing the square $
Apparently the $ AC $ automaton $ dp $
$ T3: dancing nights $
FIG built over maximum flow start and give residual network
For each edge we re-built according to the following rules map
Matching edge $ (i, j) \ \ j $ to $ I $ connected edges
mismatched edges $ (i, j) \ \ i $ to $ J $ connected side
match the left point $ i \ \ i $ to $ S $ even side
does not match the left point $ i \ \ S $ to $ I $ connected edge
matching the right point $ j \ \ T $ to $ J $ connected side
does not match the right point $ j \ \ j $ to $ T $ even edge
After this, if a non-matching edge of the start and end of a strong link in the same component, then that can have a matching side back flow to flow over him
So the final answer is that neither side match, start and end points are no longer the same point in the $ scc $