Four Mondays And Four Fridays

There are $31 = 7\times 4 + 3$ days in January. Therefore, only three days can occur $5$ times.

Neither the $1^\text{st}$ , nor the $2^\text{nd}$ , nor the $3^\text{rd}$ can be a Monday or Friday, since this would imply that either the $29^\text{th}$ , $30^\text{th}$ or $31^\text{st}$ would be the fifth Friday or Monday.

The constraints of the problem are only true when the first, second and third , respectively, are Tuesday, Wednesday and Thursday . Thus, the first of January is a Tuesday.

In a 31 days month there are:

• 3 consecutive weekdays that will appear 5 times (which are the 1st. 2nd and 3rd of the month).
• 4 consecutive weekdays that will appear 4 times,

So in this problem, those three 5 appearances days are going to be Tuesday to Thursday therefore January 1st is going to be on Tuesday.