There are 2 projective planes. Each has 2 punctures. If the PPs are joined along the 2 punctures, what will be the Euler Characteristics of the new body?
Edit 1
Each PP has two punctures.
PP1 has 2 punctures and PP2 also has 2. We now connect 1 puncture of PP1 to one of PP2 and the other puncture of PP1 to other of PP2
A projective plane with a puncture is a Möbius band, and two of them glued by their boundaries will be a Klein bottle, but the extra punctures glued apart will give you an extra handle in the Klein bottle, so your surface is $T\# K$, the genus four non orientable surface. From here you could say its Euler Characteristic.