## KingGeorge 3 years ago [SOLVED] Here's a neat little combinatorics/graph theory question. Mr. Smith and his wife invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse and nobody shook hands with the same person twice. After that, Mr. Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody. How many times did Mrs. Smith shake someone’s hand? PS. This took me about 5 minutes to solve when I was presented with it. Can you beat me?

1. ParthKohli

8 people.

2. ParthKohli

I mean, if four other couples were invited, then 4*2 = 8

3. KingGeorge

it's a little more complicated than that. Remember that everyone shook hands a different number of times (with Mr. Smith as the exception).

4. ParthKohli

Yes, I excluded Mr. Smith.

5. KingGeorge

6. KingGeorge

Barely legible hint in white LaTeX below.$\color{white}{\text{what is the maximum/minimum number of times} \text{someone could shake hands?} }$

7. Ishaan94

0? Hmm Total of 10 people and Mr. Smith shook hand with everyone right? 8 is the maximum number of handshakes? and you're asking Nine people. So, 0 for Mrs. Smith?

8. KingGeorge

It isn't 0 either. Perhaps draw a picture, and start drawing handshakes with some random person. Also, remember that he asked 9 people how many handshakes they did. What is the maximum number of handshakes a single person could have?

9. dumbcow

isn't it 0 and 8 as max/min you can't shake your own hand nor your spouse, which leaves 8 people

10. dumbcow

11. KingGeorge

That's exactly the conclusion to make. Good work so far.

12. KingGeorge

Perhaps another leading question. How many people shook hands 8 times? How many 0 times?

13. dumbcow

i believe its impossible for 2 people to have shook hands 0 or 8 times thus mr. smith shook hands somewhere between 1 and 7 following that logic, i will say 4 times is answer

14. KingGeorge

Correct. Basically, he didn't shake hands with himself or his wife. So that leaves a maximum of 8. One person shook hands 0 times, so max of 7. One person shook hands 1 time, but it had to have been with the person who shook hands 8 times. So we have a max of 6. Continue this way, and you get a max of 4. At this point, if you draw it out, 5 people have shaken hands 0, 1, 2, 3, 4 times, and 4 people have shaken hands 4, 5, 6, 7 times. To fix this, Mr. Smith has to shake hands with the last 4 people. So 4 shakes.

15. KingGeorge

Can you figure out how many times his wife shook hands?

16. dumbcow

well its not 0 for sure...kinda stuck, i think its possible she could be the one to have shaken 8 times ?

17. KingGeorge

Suppose she did shake 8 times. Who would have shaken 0 times?

18. dumbcow

oh duh, haha ok so not 8

19. dumbcow

wait i see the pattern for this to work each couple must complement each other so their sum will be 8 handshakes 0,8 1,7 2,6 3,5 4,4 Mrs. smith shook 4 times as well

20. KingGeorge

Correct again!

21. KingGeorge

I was actually going to ask "How many times did each couple shake hands next" :P Looks like you already solved that.

22. dumbcow

haha :P these are fun...takes me awhile to visualize the problem

23. KingGeorge

The way I see it for these problems, is to draw it, and to see what happens at extremes. I.e. seeing what would happen if Mr. Smith shook hands 8 times/0 times. Then try to extrapolate.