AMC 10 · 2011 · #9
Grade 5 countingPick an answer.
Every handshake has two people in it, so it is twin-with-twin, triplet-with-triplet, or twin-with-triplet, and never two of those at once. That splits one tangled count into three clean ones. Inside each single group it is easier to count all pairs and subtract the sibling pairs than to count directly. The twin-with-triplet group needs more care: the problem describes it twice, once from each side, so I read it both ways and check the two readings give the same number, then build one pattern that actually obeys both rules.
Count the people in each group
Both groups turn out to hold 18 people.
Equal groups of people mean the head count is a multiplication, not a list.
3.OA.A.1Organize Information In More WaysSort handshakes into three kinds
The greetings sort into three kinds.
Buckets that never overlap and never leave anything out let me add the bucket counts safely.
4.OA.A.3Identify SubproblemsTwin-twin: all pairs minus siblings
The first kind is all pairs minus the relatives.
Counting everything and removing the few forbidden pairs beats chasing who shakes whom.
Counting every pair and then removing the forbidden sibling pairs beats chasing who shakes whom.
▸ Why?
A handshake is one pair counted from two sides, so naming both people and halving counts it once.
▸ Why?
Every pair either shakes or is forbidden, so subtracting the forbidden ones from the whole leaves the rest.
Triplet-triplet: subtract more sibling pairs
The second removes more relative pairs.
A family of three hides three sibling pairs, not one, so bigger families remove more handshakes.
5.NBT.B.6Change Focus Count The ComplementCross handshakes, read from both sides
The cross greetings read the same from both sides.
Counting one pile from two directions is only safe when the two totals agree, so checking them is the point.
5.NF.B.4Organize Information In More WaysCheck such a convention can exist
Such an arrangement really exists.
A sliding window of nine wraps around evenly, so every triplet gets chosen by exactly nine twins.
4.OA.C.5Make A Systematic ListAdd the three kinds
Adding the kinds gives 441, choice (A).
Once the buckets do not overlap, the total is just their sum.
4.NBT.B.4Identify SubproblemsSort the handshakes by who is in them, count all pairs and subtract the ones that never happen, and remember each handshake belongs to two people but happens only once.
- Count the people in each group
- Sort handshakes into three kinds
- Twin-twin: all pairs minus siblings
- Triplet-triplet: subtract more sibling pairs
- Cross handshakes, read from both sides
- Check such a convention can exist
- Add the three kinds