# How Many Hands Did You Shake at the Party?

• MHB
• Ackbach
In summary, a handshake in this context refers to a physical greeting where two individuals grasp each other's hands and exchange a brief, friendly greeting. It is important to keep track of how many hands you shake at a party for social and contact tracing purposes. The most accurate way to count is to keep a mental or physical tally. It is not necessarily rude to decline a handshake at a party, but it is polite to offer an alternative greeting or explanation. Cultural considerations should also be taken into account, as handshakes may not be the typical form of greeting in some cultures and some individuals may have personal or religious reasons for not shaking hands.
Ackbach
Gold Member
MHB
Here is this week's Problem of the Week; you might notice a similarity with the University POTW # 147, two weeks ago, and there are. But there are also some subtle differences which completely change the solution.

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Suppose you and your husband attended a party with three other married couples. Several handshakes took place. No one shook hands with himself (or herself) or with his (or her) spouse, and no one shook hands with the same person more than once. After all the handshaking was completed, suppose you asked each person, including your husband, how many hands he or she had shaken. Each person gave a different answer. How many hands did you shake? How many hands did your husband shake?

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This problem is from Introductory Graph Theory, by Gary Chartrand.

Congratulations to Fallen Angel for his correct solution, which you can see below:

There are eight people at the party, namely $\{\text{Me},\text{My spouse},m_1, m_2, m_3 , f_1 ,f_2, f_3\}$ , where $m_i -f_i$ is a couple; each one has $6$ possibles hand-shakes and you have $7$ different answers, so the answers should be $0,1,\dots ,6$.

Assume $m_{1}$ has shaken $6$ .

Now $f_{1}$ is the only one that can shake no one (How ill-mannered! ).

If $m_{2}$ has shaken $5$, then $f_{2}$ is the only one that can shake once.

And if $m_{3}$ has shaken $4$, then $f_{3}$ has shaken twice.

So my spouse and I have shaken $3$ each.

Notice that this is totally symmetric, and every $m$ can be replaced by $f$ and vice-versa, and every $i$ by every $j$; but my spouse and I can't change with another couple because then there will be two identical answers between the guests.

## 1. How do you define a "handshake" in this context?

In this context, a handshake refers to a physical greeting where two individuals grasp each other's hands and usually exchange a brief, friendly greeting. It is a common form of social interaction in many cultures.

## 2. Why is it important to keep track of how many hands you shake at a party?

Keeping track of how many hands you shake at a party can be important for a few reasons. It can give you a sense of how many people you have interacted with and how socially active you were at the party. It can also be helpful for contact tracing in case someone at the party later tests positive for an illness.

## 3. How can you accurately count how many hands you shake at a party?

The most accurate way to count how many hands you shake at a party is to keep a mental or physical tally as you greet each person. Alternatively, you can also keep track of the number of people you interact with and assume that each person you interact with involves a handshake.

## 4. Is it rude to decline a handshake at a party?

It is not necessarily rude to decline a handshake at a party, especially if you have a valid reason such as not feeling well or if you are uncomfortable with physical contact. However, it is always polite to offer an alternative form of greeting or explanation for not shaking hands.

## 5. Are there any cultural considerations when it comes to handshakes at a party?

Yes, there are cultural considerations when it comes to handshakes at a party. In some cultures, handshakes may not be the typical form of greeting, and it is important to be aware of and respect these cultural norms. Also, some people may prefer not to shake hands for religious or personal reasons, and it is important to respect their preferences.

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