# MORE RELATED RICH PROBLEMS TO EXPLORE

- 1. Thirty people at a party shook hands with each other. How many handshakes were there altogether?
- 2. At a party, everyone shook hands with everybody else. There were 45 handshakes. How many people were at the party?
- 3. Every student in the second-grade classroom exchanged a valentine card with each other. If there were 30 students, how many valentine cards were exchanged?
- 4. Twenty people are sitting in a circle. Each person shakes hands with everyone but his/her neighbors. How many handshakes have been exchanged?
- 5. Two ten-member volleyball teams play a game. After the game, each of the members of the winning team shakes hands once with each member of both teams. How many handshakes take place?
- 6. Eight students of different heights are at a party. Each student decides to only participate in a handshake with another student shorter than himself or herself. How many handshakes take place?
- 7. Each person at a graduation party shook hands with everyone else. John shook hands with three times as many men as women. Johnâ€™s wife shook hands with four times as many men as women. How many men and women were there at the party?
- 8. Find the number of diagonals of a 12-sided polygon?
- 9. If you build a staircase with cubes that is 10 steps high, how many cubes will you need?
- 10. At a party, each man shook hands with everyone except his spouse, and no handshakes took place between women. If 13 married couples attended, how many handshakes were there among these 26 people?
- (A) 78 (B) 185 (C) 234 (D) 312 (E) 325

**Figure 1.12 Using technology to represent the worked examples of the handshake problem.**

*Source:*** Authors.**

11. (AMC8 2012) In the BIG N, a middle-school football conference, each team plays

every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

- 12. (AMC12 2011) At a twins and triplets convention, there were nine sets of twins and six sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
- (A) 324 (B) 441 (C) 630 (D)648 (E) 882