[b]p1.[/b] Can you cut the figure on the right into three congruent pieces and then put them back together to form one regular hexagon?
[img]https://cdn.artofproblemsolving.com/attachments/3/d/aee095b1cb03b98248fe1afcfcfe637d3ca033.png[/img]


[b]p2.[/b] John and Mary play in a very long chess tournament. John plays every $16$th day, while Mary plays every $25$th day. Will they sometime have to play in two consecutive days?


[b]p3.[/b] Prove that at least one of any $18$ successive three-digit numbers is divisible by the sum of its digits.


[b]p4.[/b] $30$ children from school went to a museum in pairs. After visiting the museum they went back to school also in pairs (possibly different). Show that upon their arrival back to class it is always possible to divide them into three groups such that in any group no two kids were a pair on either way.


PS. You should use hide for answers.
1. To determine if John and Mary will play on two consecutive days, we need to find the least common multiple (LCM) of the days they play. John plays every 16th day, and Mary plays every 25th day.
2. The LCM of 16 and 25 can be found by:
   \[
   \text{LCM}(16, 25) = \frac{16 \times 25}{\text{GCD}(16, 25)}
   \]
   Since 16 and 25 are coprime (their greatest common divisor, GCD, is 1), we have:
   \[
   \text{LCM}(16, 25) = 16 \times 25 = 400
   \]
3. Therefore, John and Mary will both play on the 400th day.
4. To check if they will play on consecutive days, we need to see if John plays on the 399th day or if Mary plays on the 401st day.
5. John plays every 16th day, so we check:
   \[
   399 \div 16 = 24.9375 \quad \text{(not an integer, so John does not play on the 399th day)}
   \]
6. Mary plays every 25th day, so we check:
   \[
   401 \div 25 = 16.04 \quad \text{(not an integer, so Mary does not play on the 401st day)}
   \]
7. Since neither John plays on the 399th day nor Mary plays on the 401st day, they do not play on consecutive days.

The final answer is \(\boxed{\text{No}}\)