The waiter at the restaurant U Šejdíře always adds the date to the bill: he increases the total amount spent by the paying customer by as many crowns as the day of the month it is.

In September, a trio of friends met at the restaurant twice. The first time, each of them paid separately, so the waiter added the date to each bill and asked each of them for 168 Kč. Four days later, they had lunch there again and ordered exactly the same as before. This time, however, one of them paid for everyone together. The waiter thus added the date to the bill only once and asked for 486 Kč. The friends did not think that although the prices on the menu had not changed, their lunch was cheaper than before, and they uncovered the waiter's fraud that day. What was the date?

(L. Šimünek)

Hint. Determine what their total expenditure would have been if they had each paid separately the second time as well.
If on the day when friends uncovered the waiter's fraud, each of them paid separately, the waiter would have said to each of them $168+4=172$ (CZK). In total, they would have paid $3 \cdot 172=516$ (CZK). By paying for everyone at once, the price was reduced by an amount corresponding to two dates. According to the problem, this price is 486 CZK. Therefore, the amount corresponding to two dates is $516 - 486=30$ (CZK). The number in the date is $30: 2=15$; the friends uncovered the fraud on the 15th of September.

Note. We can verify our result with a final summary: The friends first visited the restaurant on September 11th. Each of them was supposed to pay 157 CZK according to the menu, but they paid $157+11=$ $=168$ (CZK). The second time they met in the restaurant was on September 15th. One person was supposed to pay $3 \cdot 157=471$ (CZK) for everyone according to the menu, but the waiter charged $471+15=486$ (CZK).