A probability problem by Jason Chrysoprase

There are 6 pairs of distinct figures that can be extrapolated from the original shape (which contains 4 primitive pieces: triangles ARP, CRQ and quadrilaterals ADQR, BCRP).

1) Pairs that include one primitive piece: triangles ARP & CRQ + quadrilaterals ADQR & BCRP = $\boxed{2}$ total pairs;

2) Pairs that include the union of two primitive pieces: triangles ACD & ABC + trapezoids APQD & BCQP = $\boxed{2}$ total pairs;

3) Pairs that include the union of three primitive pieces: concave hexagons ADCBPR & CBADQR + concave pentagons ADCRP & ABCQR = $\boxed{2}$ total pairs;

4) Pairs that include the union of four primitive pieces: rectangle ABCD by itself = $\boxed{0}$ total pairs.

Hence, there are $\boxed{6}$ total pairs of distinct congruent figures above.