The “Handshake Problem”
□=7
□=7
□=7
□=7
□=7
□=7
□=7
□=7
8×7 = 56 handshakes
To begin, everyone read the problem and started making sense of it
for themselves. After some time, most of us came up with 56
handshakes as the answer. Then, everyone started talking to their
neighbors to see if they got the same thing. Just about everyone at one
point or another asked me if “56 handshakes” was the correct answer.
Instead of reassuring anyone, I asked people to explain their
reasoning. Jamie volunteered to come up and show the work that she
had written on her paper to help herself visualize what was going on.
She drew the work shown on the left. She explained that each box
represented one of the 8 people in the workshop, and that each of
them would shake hands with 7 other people since you can’t shake
hands with yourself. Though not everyone drew the same picture as
Jamie, her picture seemed to fit the reasoning that everyone used to
calculate the answer. After Jamie showed her solution method,
everyone agreed with her method, but there was still some uncertainty
regarding the answer. At that point, I suggested we try the solution
method with smaller numbers.
□=43
□=43
□=43
□=43
4×3 = 12 handshakes
Next, I offered a simplified version of the problem – How many
handshakes would result from a group of four people? Then I asked
someone to describe how to set up Jamie’s solution method – I was
the hand and they were the brain. At first someone said that I should
write four next to each of the boxes, but then they corrected
themselves. Once I adjusted the number of handshakes for each
person to 3, we got an answer of 12 handshakes total. People seemed
more confident, but still not positive. I suggested we act out the
problem and have four of us shake hands with each other – Jamie,
Leo, Clarice and myself all stood. As soon as we started someone
asked a really important question - if Jamie shakes hands with me,
Clarice and Leo, when it is my turn, do I shake hands with Jamie for a
second time? I responded by asking the whole group what they
thought and to look back at the problem and see what clarification
could be found. It was Leo who drew our attention to the words
“exactly once” and I asked the group what the thought of Leo’s
interpretation and what they thought the words meant in terms of the
problem. Once we all agreed that we would each only shake each
other’s hands once, we came up with six shakes total. There was still
some question about whether everyone had actually shaken hands so I
asked how we might set up a written way to keep track.
Jamie – 1
Mark – 2
Clarice – 3
Leo – 4
Jamie – 2, 3, 4
Mark – 3, 4
Clarice – 4
Leo
Our first method was to assign a number to
each of the four people shaking hands. Jamie
was 1, I was 2, Clarice was 3 and Leo was 4.
Below this we wrote each person’s name and
the number of each person they shook hands
with. This also showed six shakes that would
result, but there were some comments that it
still didn’t look clear as to who was shaking
whose hand.
Jamie
Mark
Clarice
Leo
7 shakes
Jamie
Mark
Clarice
Leo
6 shakes
Jamie – 3 shakes
Mark – 3 shakes
Clarice – 3 shakes
Leo – 3 shakes
Jane proposed our next method. As a way of
being more clear/explicit about who was
shaking hands, she told me to create two
columns, each consisting of the names of the
four people. First I was told to draw arrows
from the right to the left – from Jamie going to
me, Clarice, and Leo. Then I was told to draw
two arrows going from my name on the left to
Clarice and Leo (since I already shook Jamie’s
hand and I can’t shake my own hand). Then I
was told to draw two arrows from Clarice’s
name on the left to me and Leo. The idea was
that Clarice’s name on the left had an arrow
coming in from Jamie (on the right), so we
connected her to me and Leo. I did not say
anything about the fact that there was already a
line drawn from my name on the left to
Clarice’s name on the left. When we counted
the number of lines (each representing a
handshake), there were seven. I asked what
people thought – we came up with six and now
we were coming up with seven – What
happened? It was Jason who noticed that we
had counted the handshake between me and
Clarice twice. Once we recognized that, we
crossed out one of those lines.
We also noticed that though there are six
different handshakes, each person would shake
hands with three other people.
Jamie
Clarice
Leo
Mark
To try and make it easier to keep track of who had already shaken hands, I was told to
draw the same list of names, but only one column. Then I was told to draw arrows
down from each name for each handshake. Because we weren’t drawing any arrows
going up, we could be more confident of not duplicating our efforts. In addition to
solving the problem and further building our confidence that six was the answer, each
method addressed other, more general issues with our problem-solving strategies.
Trial (Handshake)
1
2
3
4
5
6
Initials
M+J
M+C
M+L
J+C
J+L
C+L
I asked if anyone else thought they had a way to show the situation and Jason came to
the board. His solution method also demonstrated that six was the answer. It also
showed another way to help us keep track of the handshakes – setting up a chart and
recording each shake using the initial of the two people involved. Once he was
finished, I asked how confident people were in the fact that six handshakes would
result from 4 people. Everyone felt sure, and so I drew everyone attention back to the
original problem. The answer we had was 56, but the method we used did not work
with the smaller numbers (we had gotten 12 using our first method, not the 6 we
proved to each other). Feeling confident that there were enough solution methods up
around the room, I asked each of you on your own to work on the original problem.
A
B
C
D
E
F
G
- B, C, D, E, F, G, H
- C, D, E, F, G, H
- D, E, F, G, H
- E, F, G, H
- F, G, H
- G, H
-H
H
Jamie came up to the board first and showed us how she adapted her original method.
Again, she started with a column of boxes, each representing a person, but she also
assigned each box a letter. This allowed her to keep track and make sure that each
person only shook hands one time. Once she was done, it was only a matter of
counting each letter to see that there were 28 handshakes.
Leo noticed a few things about Jamie’s method that he shared. Starting with the “H”s,
you can see that each person is shaking hands with seven other individuals. Also, there
are seven shakes in the first row, six shakes in the second row, five in the third row,
etc. Kayode also noticed that whatever letter is in the box, you could start with the
next letter of the alphabet and continue up to “H” – he called this the law of
diminishing returns.
1
2
3
4
5
6
7
8
7
6
5
4
3
2
+1
28
Leo adapted a few of the models from our group exploration of four people shaking
hands. He assigned each of the eight people a number, 1-8. In addition to drawing out
all the arrows, he also kept track of how many arrows were coming off of each person.
He also noticed that those numbers were going down by one each time, starting with
seven. He also demonstrated that there would be 28 handshakes.
Person
1
2
3
4
5
6
7
8
People They Shake Hands With
8, 7, 6, 5, 4, 3, 2
8, 7, 6, 5, 4, 3
8, 7, 6, 5, 4
8, 7, 6, 5
8, 7, 6
8, 7
8
Total Number of Handshakes
7
6
5
4
3
2
1
28
Kayode’s method had elements of both Jamie and Leo’s methods, while also making his own
sense of things. Like Leo, he assigned a number to represent each of the eight people. Also
like Leo, Kayode kept track of the total number of handshakes for each person. His center
column – “People They Shake Hands With” – works in a similar way to Jamie’s. Kayode’s
method also demonstrated that 28 handshakes would result if 8 people all shook hands, each
person shaking hands with everyone exactly once.
8–1=7
8 × 7 = 56
56 ÷ 2 = 28
4-1=3
4 × 3 = 12
12 ÷ 2 = 6
100 – 1 = 99
100 × 99 = 9900
9900 ÷ 2 = 4950
N–1
N (N-1) =
N (N-1) ÷ 2 =
Now that everyone had seen the different
visualizations, everyone was confident and
agreed that 28 was the answer. I then asked if
anyone had thought of a method that could work
for any number, especially larger numbers like
100, where some of our solution methods might
be harder. I asked this to demonstrate the point
that often working on with smaller numbers can
help us identify patterns, and make
generalizations about the relationships between
numbers. While walking around, I had seen that
Jason had started to play around with something
along these lines. I also saw that Jamie had
written “½” on her paper, perhaps noticing that
28 is half of our original guess of 56 (for 8
people) and that 6 is half of our original guess of
12 (for 4 people).
Jason said if you take the number of people and
subtract 1, you then multiply that number by the
number of people. Then you take half of that
number and you will get the number of
handshakes that would result given that number
of people. I asked him to use his procedure for
8 people and 4 people, since we were familiar
with those numbers. And sure enough his
procedure worked. Then I had someone else
take me step by step and tell me how to
determine how many handshakes would result
with 100 people, using Jason’s method.
Then I asked a few questions about how and
why Jason’s method makes sense. From the
beginning, everyone saw why we need to
subtract one from the total number of people –
you can’t shake your own hand. But why do we
divide by two?
Person
People They Shake Hands With
1
2
3
4
5
6
7
8
8, 7, 6, 5, 4, 3, 2
8, 7, 6, 5, 4, 3, 1
8, 7, 6, 5, 4, 2, 1
8, 7, 6, 5, 3, 2, 1
8, 7, 6, 4, 3, 2, 1
8, 7, 5, 4, 3, 2, 1
8, 6, 5, 4, 3, 2, 1
7, 6, 5, 4, 3, 2, 1
To try to visual why dividing by two works, I returned us to Kayode’s set-up. What
happens when we put in all of the duplicated handshakes that Kayode left out? What
do you see?
Further Food for Thought
Number of People
1
2
3
4
5
6
7
8
9
10
11
Can you complete the chart?
What pattern do you notice?
Number of Handshakes
0
1
3
6
10
15
21