connected sum of torus with projective plane

I would like to understand how to prove that the connected sum $\mathbb{R}P^2 \# T^2$ of the projective plane with a torus is homeomoprhic to $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$.

I got as far as showing that it must be equivalent to a connected sum of projective planes, how can I argue though that I need precisely three projective planes ?

Thanks for your help!

(P.S. not a homework exercise, this is for me to understand the classification of surfaces).

Solution 1:

The following picture will help you to understand this problem intuitively. (If you can understand why $\mathbb{R}\mathrm{P}^2 \# \mathbb{R}\mathrm{P}^2 = K$, where $K$ stands for the Klein bottle.)

The figure in the upper right corner is $K \setminus \text{disk} $. It may take some time to think why it looks like this.

Source of the picture: link

Solution 2:

Calculate the Euler characteristic. This, combined with the fact that the resulting surface is non-orientable gives you the complete set of invariants, enough to single out the $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$.