Let's set up variables like this: setup- There are $v_T$ dwarfs that like vanilla ice cream and tell the truth - There are $v_F$ dwarfs that like vanilla ice cream and lie - There are $c_T$ dwarfs that like chocolate ice cream and tell the truth - There are $c_F$ dwarfs that like chocolate ice cream and lie - There are $f_T$ dwarfs that like fruit ice cream and tell the truth - There are $f_F$ dwarfs that like fruit ice cream and lie A dwarf who likes a certain type of ice cream (eg chocolate) but lies will only raise their hand when the other flavors (eg vanilla and fruit) are called. We now have the system of equations: $\begin{cases} v_T+v_F+c_T+c_F+f_T+f_F = 10\\ v_T+c_F+f_F=10\\ v_F+c_T+f_F=5\\ v_F+c_F+f_T=1\\ \end{cases}$ Subtracting the second equation from the first, we have that $v_F+c_T+f_T=0$. But all of the variables can only be nonnegative (negative dwarfs, anyone?), so we must have $v_F=c_T=f_T=0$! Plugging that back in, the equations become: $\begin{cases} v_T+c_F+f_F=10\\ 0+0+f_F=5\\ 0+c_F+0=1 \end{cases} \implies v_T = 10-5-1=4$ We want all the dwarfs that tell the truth, or $v_T+c_T+f_T = 4+0+0=4$, so the answer is $\boxed{4}$ dwarfs.