Update: If you give me 175PC I'll narrow down the possible dates when I bought my Wii U to three months. Check the very first post (my post on the top of the first page) and click on the blue "Costs 175 Play Coins" button. I promise you this is legit, behind that button is your hint.
This is a good deal, allow me to treat you with a little bit of maths:
A period of four years contains 1461 days:
365+365+365+366(leap year)=1461
A period of four years contains 48 months (one year contains 12 months):
12x4=48
So therefore on average, a month contains 30.4375 days:
1461÷48=30.4375
So therefore on average, a period of three months contains 91.3125 days:
30.4375x3=91.3125
The largest number of days that you could end up guessing from is 92:
(July(31)+August(31)+September(30) OR (June(30)+July(31)+August(31)) OR (November(30)+December(31)+January(31)) OR (March(31)+April(30)+May(31)) OR (May(31)+June(30)+July(31)) OR (August(31)+September(30)+October(31)) OR (October(31)+November(30)+December(31))
The smallest number of days that you could end up guessing from is 89:
(February(28)+March(31)+April(30))
The only combinations with 90 days are:
(January(31)+February(28)+March(31)) OR (December(31)+January(31)+February(28) OR February(29)+March(31)+April(30))
All other combinations are 91 days:
(January(31)+February(29)+March(31)) OR (April(30)+May(31)+June(30) OR (September(30)+October(31)+November(30 OR December(31)+January(31)+February(29))
Anyways, imagine yourself in this situation, you've paid 175 play coins and the prize is 200PC. It costs 1PC to guess. Now you need to guess the correct answer out of 91.3125 possible dates in under 25 guesses in order to win, if you go over that however, you will lose, and if takes you 25 guesses to guess the answer, then you will tie.
So the chance of you winning after paying 175PC is 24/91.3125. That is slightly more than 26%. Compare that with a chance of 199/2462 which is slightly more than 8%.
Your chances of winning increase by more than 300% if you buy the hint!!