In a certain year, January had exactly four Tuesdays and four Saturdays.  On what day did January $1$ fall that year?
$\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Tuesday} \qquad \text{(C)}\ \text{Wednesday} \qquad \text{(D)}\ \text{Friday} \qquad \text{(E)}\ \text{Saturday}$
January has four full weeks and then three extra consecutive days. Each full week contributes one Tuesday and one Saturday, so the three extra days do not contain a Tuesday and Saturday.  Therefore, those three days are Wednesday, Thursday, and Friday. 
Wednesday is the 29th day of January, therefore the 22nd, 15th, 8th, and 1st of January are all Wednesdays, so the answer is $\boxed{\text{C}}$