Problem
Count handshakes in a group
Every club member shakes hands with every other member. Each pair shakes hands only once. The book club has 5 people and the dance club 6. Count the handshakes in each club.
Reasoning · Grade 4-2 Leagues and Tournaments
Your answer
Watch out Four each for five people gives 5 × 4, but that counts each handshake twice.
How to solve
Strategy Draw a Diagram — Names are hard to keep track of, so I turn each person into a dot on a circle and each handshake into a line joining two dots. Now the question is simply how many lines there are, which I can literally see and count. A systematic list - go round the members and count only the handshakes not already recorded - makes the count airtight for 5 people. That list then reveals a multiply-then-halve pattern, which handles the 6-person club instantly and doubles as a check on the first answer.
1STEP 1
Turn people into dots and handshakes into lines
Draw people as dots and handshakes as lines.
5 people → 5 dots
A handshake connects exactly two people, and a line segment connects exactly two dots, so the picture stores the information perfectly while being far easier to count.
4.G.A.1Draw A Diagram2STEP 2
List the book club's handshakes without repeats
Without repeats, 4 + 3 + 2 + 1 = 10.
4 + 3 + 2 + 1 = 10
Counting only what is new at each step is what keeps the same handshake from being counted twice, and the falling 4, 3, 2, 1 shows the double counting being squeezed out one person at a time.
4.OA.A.3Make A Systematic List3STEP 3
Find the multiply-then-halve shortcut
Halving 5 × 4 also gives 10.
5 × 4 = 20, 20 ÷ 2 = 10
Two hands meet in every handshake, so asking everyone how many they made always gives exactly double the truth - halving is the honest correction.
3.OA.A.3Look For A PatternAsking everyone how many hands they shook gives exactly double the true number of handshakes, so halving corrects it.
Why?
Two hands meet in every handshake, so each handshake is reported once by each of the two people in it.
🧱Counted twice, so halveIf every item got counted from both of its two ends, the total is exactly double the truth, so halving fixes it.
Why?
Every person can shake with every other person and nobody is restricted, so the reported total is one count multiplied by another.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
4STEP 4
Do the dance club the same way
For 6 people, 6 × 5 ÷ 2 = 15.
6 × 5 ÷ 2 = 15 and 5+4+3+2+1 = 15
Two completely different countings landing on 15 is much better evidence than one counting done twice.
3.OA.A.3Look For A Pattern5STEP 5
Check the two answers against each other
One extra person adds 5 handshakes — it checks.
10 + 5 = 15
If the two answers had not differed by exactly 5, one of them would have to be wrong - so this link is a real test, not just a restatement.
4.OA.C.5Look For A PatternAnswer
10, 15 handshakes
5 × 4 ÷ 2 = 10, 6 × 5 ÷ 2 = 15
The answers are counts of handshakes, so they must be whole numbers, and they are. Each has to be at least the number of people (everyone makes at least one) and well below 5 x 4 = 20 and 6 x 5 = 30, which would be the totals if every handshake were counted from both sides - and 10 and 15 are exactly half of those, as they should be. For the 5-person club the picture can be checked by eye: a ring of 5 dots joined every which way is a pentagon with a star inside, which is 5 sides plus 5 diagonals, 10 lines in all.
Takeaway
Every handshake gets counted twice - once by each hand - so multiply people by one-less-than-people, then halve.
- Turn people into dots and handshakes into lines
- List the book club's handshakes without repeats
- Find the multiply-then-halve shortcut
- Do the dance club the same way
- Check the two answers against each other